Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2013 Article ID 450481 10 pageshttpdxdoiorg1011552013450481

Research Article119902-Pascal and 119902-Wronskian Matrices with Implications to119902-Appell Polynomials

Thomas Ernst

Department of Mathematics Uppsala University PO Box 480 751 06 Uppsala Sweden

Correspondence should be addressed toThomas Ernst thomasmathuuse

Received 19 June 2012 Accepted 2 December 2012

Academic Editor Franck Petit

Copyright copy 2013 Thomas ErnstThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We introduce a 119902-deformation of the Yang andYounmatrix approach forAppell polynomialsThiswill lead to a powerfulmachineryfor producing new and old formulas for 119902-Appell polynomials and in particular for 119902-Bernoulli and 119902-Euler polynomialsFurthermore the 119902-119871-polynomial anticipated by Ward can be expressed as a sum of products of 119902-Bernoulli and 119902-Eulerpolynomials The pseudo 119902-Appell polynomials which are first presented in this paper enable multiple 119902-analogues of the Yangand Youn formulas The generalized 119902-Pascal functional matrix the 119902-Wronskian vector of a function and the vector of 119902-Appellpolynomials together with the 119902-deformed 120598 matrix multiplication from the authors recent article are the main ingredients in theprocess Beyond these results we give a characterization of 119902-Appell numbers improving on Al-Salam 1967 Finally we find a119902-difference equation for the 119902-Appell polynomial of degree 119899

1 Introduction

In this paper we will introduce the 119902-Pascal and 119902-Wronskianmatrices in a general setting with the aim of fruitful applica-tions for 119902-Appell polynomials These two matrices containa certain 119902-difference operator just like the 119902-deformedLeibniz functional matrix from [1] By the 119902-Leibniz formulaproducts of suchmatrices contain a certain operator sdot

120598 just as

in the previous article but with a slightly different definitionHowever since in almost all equations we compute thefunction value at 119905 = 0 this operator will convert to ordinarymatrixmultiplication by formula (6)There is a connection tothe 119902-Pascal matrix [2] for the special case 119891 = 119902-exponentialfunctionThe Appell polynomials are seldom encountered inthe literature one exception was Carlitzrsquo article [3] wherea formula for the product of two generating functions wasgiven Carlitz cleverly observed that the generating functionfor Appell polynomials can be inverted when 119891(0) = 1 TheNWA 119902-addition seldom occurs in the literature except forthe authorrsquos papers it was first seen in an Italian paper byNalli[4] Corresponding to the two dual 119902-additions NWA andJHC there are two dual 119902-Bernoulli polynomials with thesame names these are special cases of 119902-Appell polynomialsthe content of Section 2 It turns out that for certain purposes

wewill need pseudo 119902-Appell polynomials which correspondto the other basic 119902-addition JHC these polynomials werefirst seen in [5] The 119902-Appell numbers form a certainalgebraic structure with two operations which was firstdescribed by Al-Salam [6] This structure will in a certainway be generalized to 119902-Appell polynomials The 119902-Bernoulliand 119902-Euler polynomials are presented along with the 119902-Lucas polynomials these definitions coincide with thosegiven in [7] In Section 3 we come to the core of the paperwhich starts with the 119902-Pascal matrix and the 119902-Wronskianvector In the beginning we briefly pass to the shorter paper[8] to find a few 119902-analogues of truncated 119902-exponentialgenerating function formulas We mostly follow the orderingin [9] Sometimes multiple 119902-analogues follow and thesewill usually come with the pseudo 119902-Appell polynomialsA couple of general formulas for 119902-Appell polynomials aregiven sometimes these are specialized to 119902-Bernoulli and 119902-Euler polynomials In the last part we find a 119902-differenceequation for general 119902-Appell polynomials In an appendixwecompare our formulas for Bernoulli and Euler polynomialswith Hansenrsquos table [10] where another notation is used

We now start with the definition of the umbral methodfrom the recent book [11] Some of the notations are wellknown and will be skipped

2 Journal of Discrete Mathematics

Definition 1 Let the Gauss 119902-binomial coefficient be definedby

(119899

119896)

119902

equiv119899119902

119896119902119899 minus 119896

119902 119896 = 0 1 119899 (1)

Let 119886 and 119887 be any elements with commutativemultiplicationThen the NWA 119902-addition is given by

(119886 oplus119902119887)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

119886119896

119887119899minus119896

119899 = 0 1 2 (2)

The JHC 119902-addition is the function

(119886 ⊞119902119887)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

119902(119896

2)

119887119896

119886119899minus119896

= 119886119899

(minus119887

119886 119902)

119899

119899 = 0 1 2

(3)

The 119902-exponential function is defined by

E119902(119911) equiv

infin

sum

119896=0

1

119896119902119911119896

(4)

The 119902-derivative is defined by

(D119902119891) (119909) equiv

119891 (119909) minus 119891 (119902119909)

(1 minus 119902) 119909 if 119902 isin C 1 119909 = 0

119889119891

119889119909(119909) if 119902 = 1

119889119891

119889119909(0) if 119909 = 0

(5)

Assume that 119905119910 and 119902 are the variables If wewant to indicatethe variable on which the 119902-difference operator is applied wedenote the operator in (5) by (D

119902119905120593)(119905 119910) To save space we

sometimes write 1198911015840 instead of D119902119905119891

Theorem2 The following equation relates the 119899th 119902-differenceoperator at zero to the 119899th derivative at zero for an analyticfunction 119891(119911) [12]

D119899119902119891 (0) =

119899119902

119899119891(119899)

(0) (6)

For convenience we will use the following two abbrevia-tions for functions 119891(119905) 119891(119896) represents the 119896th 119902-derivativeof 119891 and 119891

119896 represents the 119896th power of 119891 We will onlyconsider functions in C[[119905]] Sometimes functions can begiven in other forms we then mean the Taylor expansion ofthe function around zero In the sense ofMilne-Thomson [13]and Rainville [14] we write = to indicate the symbolic nature

If 119891(119909) isin C[119909] the ring of polynomials with complexcoefficients then the function 120598 C[119909] 997891rarr C[119909] is definedby

120598119891 (119909) equiv 119891 (119902119909) (7)

We make the convention that all 119899 times 119899 matrices are denotedby the first index 119899 The rows and columns are denoted by 119894and 119895 The 119902-Pascal matrix [2] is defined by

119875119899119902(119909) (119894 119895) equiv (

119894

119895)

119902

119909119894minus119895

(8)

The Polya-Vein matrix119867119899119902

[2] is given by

119867119899119902(119894 119894 minus 1) equiv 119894

119902 119894 = 1 119899 minus 1

119867119899119902(119894 119895) equiv 0 119895 = 119894 minus 1

(9)

2 119902-Appell Polynomials

The Appell polynomials have been around for more than acentury but this general notion is still not so well knownTheusual definition is

119891 (119905) 119890119909119905

=

infin

sum

120584=0

119905120584

120584Φ120584(119909) 119891 (119905) isin C [[119909]] (10)

The original article by Appell [15] started with the definition119889119860119899119889119909 = 119899119860

119899minus1 Then Appell showed that multiplication of

two Appell polynomials is commutative He then consideredthe inverse of119860

119899as a natural generalization of ordinary divi-

sionThe 119902-Appell polynomials form a natural generalizationof the Appell polynomials

21 Definitions Weconsider a certain value of the order (= 1)for the 119902-Appell polynomials in [11] The degree is only aname it is not the degree of polynomials

Definition 3 The 119902-Appell polynomials Φ120584119902(119909) of degree 120584

are defined by

119891 (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Φ120584119902(119909) 119891 (119905) isin C [[119909]] (11)

We say that Φ120584119902(119909) is the 119902-Appell polynomial sequence for

119891(119905)We put 119909 = 0 and obtain

119891 (119905) =

infin

sum

120584=0

119905120584

120584119902Φ120584119902 Φ

0119902= 1 (12)

where Φ120584119902

is said to have degree 120584

Definition 4 The pseudo 119902-Appell polynomials Φ119902

120584(119909) of

degree 120584 [5] are defined by

119891 (119905)E1119902

(119909119905) =

infin

sum

120584=0

119905120584

120584119902Φ119902

120584(119909) (13)

We say that Φ119902120584(119909) is the pseudo 119902-Appell polynomial seq-

uence for 119891(119905)By putting 119909 = 0 we have

119891 (119905) =

infin

sum

120584=0

119905120584

120584119902Φ119902

120584 (14)

where Φ119902120584is called a Φ119902 number of degree 120584

Journal of Discrete Mathematics 3

We have similarly

D119902Φ119902

120584(119909) = 120584

119902Φ119902

120584minus1119902(119902119909)

Φ119902

120584(119909) = (Φ

119902⊞119902119909)120584

(15)

22 119902-Analogue of Carlitzrsquo Product Formula (1963) To moti-vate the generalized 119902-Pascal functional matrix we show thatthe Carlitz [3 page 247] formula for the product of generatingfunctions for Appell polynomials has a 119902-analogue involvingthe NWA 119902-addition Define the numbers Φ1015840

120584119902infin

120584=0by the

inverse of (12) (this is allowable since 119891(0) = 1)

119891(119905)minus1

=

infin

sum

120584=0

119905120584

120584119902Φ1015840

120584119902 Φ1015840

0119902= 1 (16)

Then we have the following

Theorem 5 An inverse relation formula for the 119902-Appellpolynomials

119909119899

=

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902Φ119896119902(119909) (17)

Proof By the umbral definition of 119902-Appell polynomials wehave

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902Φ119896119902(119909)

=

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902

119896

sum

119897=0

(119896

119897)

119902

Φ119896minus119897119902

119909119897

=

119899

sum

119896=0

119899119902Φ1015840

119899minus119896119902

119899 minus 119896119902

119899

sum

119897=0

Φ119896minus119897119902

119897119902119896 minus 119897

119902119909119897

=

119899

sum

119897=0

119899119902

119897119902119909119897

sum

119904+119905=119899minus119897

Φ1015840

119904119902Φ119905119902

119904119902119905119902

=

119899

sum

119897=0

119899119902

119897119902119909119897

120575119899minus119897

= 119909119899

(18)

This completes the proof

We introduce two other 119902-Appell polynomials Ξ119902and Ψ

119902

by

119892 (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Ξ120584119902(119909) 119892 (119905) isin C [[119909]]

119892 (119905) =

infin

sum

120584=0

119905120584

120584119902Ξ120584119902 Ξ0119902

= 1

ℎ (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Ψ120584119902(119909) ℎ (119905) isin C [[119909]]

ℎ (119905) =

infin

sum

120584=0

119905120584

120584119902Ψ120584119902 Ψ0119902

= 1

(19)

Let us consider119891119898(119886119909) 119892

119899(119887119909)

=

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

119909119903+119904

(20)

Now put

ℎ(119905)minus1

=

infin

sum

120584=0

119905120584

120584119902Ψ1015840

120584119902 (21)

Because of (17) we get

119891119898(119886119909) 119892

119899(119887119909)

=

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

119903+119904

sum

119895=0

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902Ψ119895119902(119909)

(22)

Introduce the notation119860 (119895 119902119898 119899 119886 119887)

equiv

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

(119903 + 119904

119895)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

Ψ1015840

119903+119904minus119895119902

(23)

where

119891119898(119886119909) 119892

119899(119887119909) =

119898+119899

sum

119895=0

119860 (119895 119902119898 119899 119886 119887)Ψ119895119902(119909) (24)

Theorem 6 With the previously mentioned notation one hasthe following 119902-analogue of Carlitz [3 page 247]

infin

sum

119898=0

infin

sum

119899=0

119860 (119895 119902119898 119899 119886 119887)119906119898

119907119899

119898119902119899119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=0

Ψ1015840

119896119902

(119886119906 oplus119902119887119907)119896+119895

119895119902119896119902

(25)

Proof Thus we haveinfin

sum

119898=0

infin

sum

119899=0

119860 (119895 119902119898 119899 119886 119887)119906119898

119907119899

119898119902119899119902

=

infin

sum

119898=0

infin

sum

119899=0

119906119898

119907119899

119898119902119899119902

119898

sum

119903=0

119899

sum

119904=0

= (119898

119903)

119902

(119899

119904)

119902

(119903 + 119904

119895)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

Ψ1015840

119903+119904minus119895119902

=

infin

sum

119903=0

infin

sum

119904=0

(119886119906)119903

(119887119907)119904

119903119902119904119902

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

infin

sum

119898=0

infin

sum

119899=0

119891119898119892119899

119906119898

119907119899

119898119902119899119902

= 119891 (119906) 119892 (119907)

infin

sum

119903=0

infin

sum

119904=0

(119886119906)119903

(119887119907)119904

119903119902119904119902

4 Journal of Discrete Mathematics

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902sum

119903+119904=119896

(119886119906)119903

(119887119907)119904

119903119902119904119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902

(119886119906 oplus119902119887119907)119896

119896119902

(26)

23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]

Definition 7 We denote by A119902the set of all 119902-Appell-

numbersLet Φ

119899119902and Ψ

119899119902be two elements in A

119902 Then the

operations + and ⋆ are defined as follows

(Φ119902+ Ψ119902)119899

equiv Φ119899119902+ Ψ119899119902 (27)

(Φ119902⋆ Ψ119902)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

Φ119899minus119896119902

Ψ119896119902 (28)

Theorem 8 The identity element in A119902with respect to ⋆ is

119868 equiv 1205751198990 This corresponds to 119891(119905) = 1

Proof Put 119896 = 119899 and Φ119902= 1 in (28)

Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902

andΨ119899119902

be three elements in A119902corresponding to the generating

functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then

Φ119899119902+ Ψ119899119902

isin A119902

Φ119899119902+ Ψ119899119902

has generating function119891 (119905) + ℎ (119905) (29)

The associative law for addition reads

Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ

119899119902+ Ξ119899119902) + Ψ119899119902 (30)

Proof This follows from the definitions

Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and

Ψ119899119902

be three elements in A119902corresponding to the generating

functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then

Φ119899119902⋆ Ψ119899119902

isin A119902 (31)

Φ119899119902⋆ Ψ119899119902

= Ψ119899119902⋆ Φ119899119902 (32)

Φ119899119902⋆ Ψ119899119902

has generating function119891 (119905) ℎ (119905) (33)

The associative law for ⋆ reads

Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ

119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)

Proof We prove (31) and (33) at the same time We have inthe umbral sense

119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)

Formulas (32) and (34) follow by the commutativity andthe associativity of NWA

24 Specializations

Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by

1199051

(E119902(119905) minus 1)

E119902(119909119905) =

infin

sum

120584=0

119905120584BNWA120584119902 (119909)

120584119902

|119905| lt 2120587 (36)

The NWA 119902-Bernoulli polynomials are also given by

BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584

(37)

The pseudo-119902-Bernoulli polynomials are defined by

B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584

(38)

The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by

1199051

(E1119902

(119905) minus 1)E119902(119909119905) =

infin

sum

120584=0

119905120584BJHC120584119902 (119909)

120584119902

|119905| lt 2120587 (39)

3 119902-Pascal and 119902-Wronskian Matrices

We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread

Definition 12 The generalized 119902-Pascal functional matrix isgiven by

(P119899119902) [119891 (119905 119902)] (119894 119895)

equiv

(119894

119895)

119902

D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(40)

In the special case 119891(119905 119902) = constant

(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)

Definition 13 The 119902-deformed Wronskian vector is given by

(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894

119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1

(42)

Journal of Discrete Mathematics 5

Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)

equiv

D119894minus119895119902119905119891 (119905 119902)

119894 minus 119895119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(43)

In order to find a 119902-analogue of the formula [16]

(L119899) [119891 (119905) 119892 (119905)] = (L

119899) [119891 (119905)] (L

119899) [119892 (119905)] (44)

we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]

= (L119899119902) [119891 (119905 119902)] sdot

120598(L119899119902) [119892 (119905 119902)]

(45)

where in the matrix multiplication for every term whichincludes D119896

119902119891 we operate with 120598119896 on 119892 We denote this by sdot

120598

This operator can also be iterated compare with [1]

Theorem 15 (a 119902-analogue of [9 page 67]) Consider

P119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)

where the 119902-Pascal matrix is defined in (8)The mappingsP

119899119902[sdot] andW

119899119902[sdot] are linear that is

P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P

119899119902[119891 (119905)] + 119887P

119899119902[119892 (119905)]

W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W

119899119902[119891 (119905)] + 119887W

119899119902[119892 (119905)]

(47)

One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]

P119899119902[119891 (119905 119902)] sdot

120598P119899119902[119892 (119905 119902)] = P

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(48)

P119899119902[119891 (119905 119902)] sdot

120598W119899119902[119892 (119905 119902)] = W

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(49)

The scaling properties can be stated as follows

W119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

(50)

P119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

times diag [1 119886minus1 1198861minus119899]

(51)

Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule

Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as

P119899119902[119891 (119905 119902)]

minus1119902

equiv P119899119902[119891(119905 119902)

minus1

] (52)

Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let

(G119899119902) [119909] (119894 119895)

equiv

119892119894minus119895(119909) (

119894

119895)

119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(53)

Put

119891 (119905 119909 119902) equiv

119899

sum

119896=0

119892119896(119909)

119905119896

119896119902 (54)

a truncated 119902-exponential generating function of 119902-binomialtype We find that

P119899119902[119891 (119905 119909 119902)]

1003816100381610038161003816119905=0 = G119899119902[119909] (55)

By a routine computation we obtain

G119899119902[119909]G119899119902[119910] = G

119899119902[119909 oplus119902119910] (56)

The function argument 119909oplus119902119910 is induced from the corre-

sponding 119902-additionIn the same way we obtain

G119899119902[119909]G1198991119902

[119910] = G119899119902[119909 ⊞119902119910] (57)

By the last formula we can define the inverse ofG119899119902[119909] as

G119899119902[119909]minus1

equiv G1198991119902

[minus119909] (58)

We now return permanently to [9]

Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-

Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ

119902(119909) for 119891(119905) is given by

Φ119902(119909) (119894) equiv Φ

119894119902(119909) 119894 = 0 1 119899 minus 1 (59)

Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899

(D119902119909) 119910 (119909) = P

119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin

(60)

with initial value 119910(0) = 1199100 has the following solution

119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]

10038161003816100381610038161003816119905=01199100 (61)

Proof Since E119902(119860119905)|119905=0

is equal to the unit matrix the solu-tion of (60) is

Φ119902(119909) = E

119902(P119899119902[119891 (119905)]

1003816100381610038161003816119905=0119909) 1199100

= (

infin

sum

119896=0

P119896

119899119902[119891 (119905)])

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

119909119896

119896119902

(62)

6 Journal of Discrete Mathematics

By (47) and (48)

Φ119902(119909) = P

119899119902(

infin

sum

119896=0

119891119896

(119905)119909119896

119896119902)

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

(63)

where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot

120598

Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if

Φ (119909) = P119899119902[119891 (119905)]

1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(64)

Proof We know from [2 17] thatP119899119902[119905]|119905=0

= 119867119899119902 (65)

We thus obtainΦ (119909) = P

119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]

1003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(66)

Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by

P119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0 (67)

Can be compared with (53)

Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)

Ξ120584119902(119909) andΨ

120584119902(119909) be the 119902-Appell polynomials corresponding

to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then

Ξ120584119902(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119896119902(119911) (68)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)] sdot

120598W119899119902[ℎ (119905)E

119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 oplus119902119911) 119905)]

10038161003816100381610038161003816119905=0

(69)

By (11) the left-hand side of (69) is equal to

(((((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))))

)

times(

Ψ0119902(119911)

Ψ1119902(119911)

Ψ119899minus1119902

(119911)

)

(70)

By (11) the right-hand side of (69) is equal to

(

Ξ0119902(119910 oplus119902119911)

Ξ1119902(119910 oplus119902119911)

Ξ119899minus1119902

(119910 oplus119902119911)

) (71)

The theorem follows by equating the last rows of formulas (71)and (70)

Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which

enables inverses

Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902

and Ξ120584119902

be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902

120584be the pseudo 119902-Appell polynomial

for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then

Ξ120584119902(119910 ⊞119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119902

119896(119911) (72)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)]

10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E

1119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 ⊞119902119911) 119905)] |

119905=0

(73)

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Discrete Mathematics

Definition 1 Let the Gauss 119902-binomial coefficient be definedby

(119899

119896)

119902

equiv119899119902

119896119902119899 minus 119896

119902 119896 = 0 1 119899 (1)

Let 119886 and 119887 be any elements with commutativemultiplicationThen the NWA 119902-addition is given by

(119886 oplus119902119887)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

119886119896

119887119899minus119896

119899 = 0 1 2 (2)

The JHC 119902-addition is the function

(119886 ⊞119902119887)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

119902(119896

2)

119887119896

119886119899minus119896

= 119886119899

(minus119887

119886 119902)

119899

119899 = 0 1 2

(3)

The 119902-exponential function is defined by

E119902(119911) equiv

infin

sum

119896=0

1

119896119902119911119896

(4)

The 119902-derivative is defined by

(D119902119891) (119909) equiv

119891 (119909) minus 119891 (119902119909)

(1 minus 119902) 119909 if 119902 isin C 1 119909 = 0

119889119891

119889119909(119909) if 119902 = 1

119889119891

119889119909(0) if 119909 = 0

(5)

Assume that 119905119910 and 119902 are the variables If wewant to indicatethe variable on which the 119902-difference operator is applied wedenote the operator in (5) by (D

119902119905120593)(119905 119910) To save space we

sometimes write 1198911015840 instead of D119902119905119891

Theorem2 The following equation relates the 119899th 119902-differenceoperator at zero to the 119899th derivative at zero for an analyticfunction 119891(119911) [12]

D119899119902119891 (0) =

119899119902

119899119891(119899)

(0) (6)

For convenience we will use the following two abbrevia-tions for functions 119891(119905) 119891(119896) represents the 119896th 119902-derivativeof 119891 and 119891

119896 represents the 119896th power of 119891 We will onlyconsider functions in C[[119905]] Sometimes functions can begiven in other forms we then mean the Taylor expansion ofthe function around zero In the sense ofMilne-Thomson [13]and Rainville [14] we write = to indicate the symbolic nature

If 119891(119909) isin C[119909] the ring of polynomials with complexcoefficients then the function 120598 C[119909] 997891rarr C[119909] is definedby

120598119891 (119909) equiv 119891 (119902119909) (7)

We make the convention that all 119899 times 119899 matrices are denotedby the first index 119899 The rows and columns are denoted by 119894and 119895 The 119902-Pascal matrix [2] is defined by

119875119899119902(119909) (119894 119895) equiv (

119894

119895)

119902

119909119894minus119895

(8)

The Polya-Vein matrix119867119899119902

[2] is given by

119867119899119902(119894 119894 minus 1) equiv 119894

119902 119894 = 1 119899 minus 1

119867119899119902(119894 119895) equiv 0 119895 = 119894 minus 1

(9)

2 119902-Appell Polynomials

The Appell polynomials have been around for more than acentury but this general notion is still not so well knownTheusual definition is

119891 (119905) 119890119909119905

=

infin

sum

120584=0

119905120584

120584Φ120584(119909) 119891 (119905) isin C [[119909]] (10)

The original article by Appell [15] started with the definition119889119860119899119889119909 = 119899119860

119899minus1 Then Appell showed that multiplication of

two Appell polynomials is commutative He then consideredthe inverse of119860

119899as a natural generalization of ordinary divi-

sionThe 119902-Appell polynomials form a natural generalizationof the Appell polynomials

21 Definitions Weconsider a certain value of the order (= 1)for the 119902-Appell polynomials in [11] The degree is only aname it is not the degree of polynomials

Definition 3 The 119902-Appell polynomials Φ120584119902(119909) of degree 120584

are defined by

119891 (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Φ120584119902(119909) 119891 (119905) isin C [[119909]] (11)

We say that Φ120584119902(119909) is the 119902-Appell polynomial sequence for

119891(119905)We put 119909 = 0 and obtain

119891 (119905) =

infin

sum

120584=0

119905120584

120584119902Φ120584119902 Φ

0119902= 1 (12)

where Φ120584119902

is said to have degree 120584

Definition 4 The pseudo 119902-Appell polynomials Φ119902

120584(119909) of

degree 120584 [5] are defined by

119891 (119905)E1119902

(119909119905) =

infin

sum

120584=0

119905120584

120584119902Φ119902

120584(119909) (13)

We say that Φ119902120584(119909) is the pseudo 119902-Appell polynomial seq-

uence for 119891(119905)By putting 119909 = 0 we have

119891 (119905) =

infin

sum

120584=0

119905120584

120584119902Φ119902

120584 (14)

where Φ119902120584is called a Φ119902 number of degree 120584

Journal of Discrete Mathematics 3

We have similarly

D119902Φ119902

120584(119909) = 120584

119902Φ119902

120584minus1119902(119902119909)

Φ119902

120584(119909) = (Φ

119902⊞119902119909)120584

(15)

22 119902-Analogue of Carlitzrsquo Product Formula (1963) To moti-vate the generalized 119902-Pascal functional matrix we show thatthe Carlitz [3 page 247] formula for the product of generatingfunctions for Appell polynomials has a 119902-analogue involvingthe NWA 119902-addition Define the numbers Φ1015840

120584119902infin

120584=0by the

inverse of (12) (this is allowable since 119891(0) = 1)

119891(119905)minus1

=

infin

sum

120584=0

119905120584

120584119902Φ1015840

120584119902 Φ1015840

0119902= 1 (16)

Then we have the following

Theorem 5 An inverse relation formula for the 119902-Appellpolynomials

119909119899

=

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902Φ119896119902(119909) (17)

Proof By the umbral definition of 119902-Appell polynomials wehave

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902Φ119896119902(119909)

=

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902

119896

sum

119897=0

(119896

119897)

119902

Φ119896minus119897119902

119909119897

=

119899

sum

119896=0

119899119902Φ1015840

119899minus119896119902

119899 minus 119896119902

119899

sum

119897=0

Φ119896minus119897119902

119897119902119896 minus 119897

119902119909119897

=

119899

sum

119897=0

119899119902

119897119902119909119897

sum

119904+119905=119899minus119897

Φ1015840

119904119902Φ119905119902

119904119902119905119902

=

119899

sum

119897=0

119899119902

119897119902119909119897

120575119899minus119897

= 119909119899

(18)

This completes the proof

We introduce two other 119902-Appell polynomials Ξ119902and Ψ

119902

by

119892 (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Ξ120584119902(119909) 119892 (119905) isin C [[119909]]

119892 (119905) =

infin

sum

120584=0

119905120584

120584119902Ξ120584119902 Ξ0119902

= 1

ℎ (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Ψ120584119902(119909) ℎ (119905) isin C [[119909]]

ℎ (119905) =

infin

sum

120584=0

119905120584

120584119902Ψ120584119902 Ψ0119902

= 1

(19)

Let us consider119891119898(119886119909) 119892

119899(119887119909)

=

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

119909119903+119904

(20)

Now put

ℎ(119905)minus1

=

infin

sum

120584=0

119905120584

120584119902Ψ1015840

120584119902 (21)

Because of (17) we get

119891119898(119886119909) 119892

119899(119887119909)

=

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

119903+119904

sum

119895=0

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902Ψ119895119902(119909)

(22)

Introduce the notation119860 (119895 119902119898 119899 119886 119887)

equiv

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

(119903 + 119904

119895)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

Ψ1015840

119903+119904minus119895119902

(23)

where

119891119898(119886119909) 119892

119899(119887119909) =

119898+119899

sum

119895=0

119860 (119895 119902119898 119899 119886 119887)Ψ119895119902(119909) (24)

Theorem 6 With the previously mentioned notation one hasthe following 119902-analogue of Carlitz [3 page 247]

infin

sum

119898=0

infin

sum

119899=0

119860 (119895 119902119898 119899 119886 119887)119906119898

119907119899

119898119902119899119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=0

Ψ1015840

119896119902

(119886119906 oplus119902119887119907)119896+119895

119895119902119896119902

(25)

Proof Thus we haveinfin

sum

119898=0

infin

sum

119899=0

119860 (119895 119902119898 119899 119886 119887)119906119898

119907119899

119898119902119899119902

=

infin

sum

119898=0

infin

sum

119899=0

119906119898

119907119899

119898119902119899119902

119898

sum

119903=0

119899

sum

119904=0

= (119898

119903)

119902

(119899

119904)

119902

(119903 + 119904

119895)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

Ψ1015840

119903+119904minus119895119902

=

infin

sum

119903=0

infin

sum

119904=0

(119886119906)119903

(119887119907)119904

119903119902119904119902

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

infin

sum

119898=0

infin

sum

119899=0

119891119898119892119899

119906119898

119907119899

119898119902119899119902

= 119891 (119906) 119892 (119907)

infin

sum

119903=0

infin

sum

119904=0

(119886119906)119903

(119887119907)119904

119903119902119904119902

4 Journal of Discrete Mathematics

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902sum

119903+119904=119896

(119886119906)119903

(119887119907)119904

119903119902119904119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902

(119886119906 oplus119902119887119907)119896

119896119902

(26)

23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]

Definition 7 We denote by A119902the set of all 119902-Appell-

numbersLet Φ

119899119902and Ψ

119899119902be two elements in A

119902 Then the

operations + and ⋆ are defined as follows

(Φ119902+ Ψ119902)119899

equiv Φ119899119902+ Ψ119899119902 (27)

(Φ119902⋆ Ψ119902)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

Φ119899minus119896119902

Ψ119896119902 (28)

Theorem 8 The identity element in A119902with respect to ⋆ is

119868 equiv 1205751198990 This corresponds to 119891(119905) = 1

Proof Put 119896 = 119899 and Φ119902= 1 in (28)

Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902

andΨ119899119902

be three elements in A119902corresponding to the generating

functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then

Φ119899119902+ Ψ119899119902

isin A119902

Φ119899119902+ Ψ119899119902

has generating function119891 (119905) + ℎ (119905) (29)

The associative law for addition reads

Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ

119899119902+ Ξ119899119902) + Ψ119899119902 (30)

Proof This follows from the definitions

Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and

Ψ119899119902

be three elements in A119902corresponding to the generating

functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then

Φ119899119902⋆ Ψ119899119902

isin A119902 (31)

Φ119899119902⋆ Ψ119899119902

= Ψ119899119902⋆ Φ119899119902 (32)

Φ119899119902⋆ Ψ119899119902

has generating function119891 (119905) ℎ (119905) (33)

The associative law for ⋆ reads

Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ

119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)

Proof We prove (31) and (33) at the same time We have inthe umbral sense

119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)

Formulas (32) and (34) follow by the commutativity andthe associativity of NWA

24 Specializations

Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by

1199051

(E119902(119905) minus 1)

E119902(119909119905) =

infin

sum

120584=0

119905120584BNWA120584119902 (119909)

120584119902

|119905| lt 2120587 (36)

The NWA 119902-Bernoulli polynomials are also given by

BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584

(37)

The pseudo-119902-Bernoulli polynomials are defined by

B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584

(38)

The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by

1199051

(E1119902

(119905) minus 1)E119902(119909119905) =

infin

sum

120584=0

119905120584BJHC120584119902 (119909)

120584119902

|119905| lt 2120587 (39)

3 119902-Pascal and 119902-Wronskian Matrices

We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread

Definition 12 The generalized 119902-Pascal functional matrix isgiven by

(P119899119902) [119891 (119905 119902)] (119894 119895)

equiv

(119894

119895)

119902

D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(40)

In the special case 119891(119905 119902) = constant

(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)

Definition 13 The 119902-deformed Wronskian vector is given by

(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894

119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1

(42)

Journal of Discrete Mathematics 5

Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)

equiv

D119894minus119895119902119905119891 (119905 119902)

119894 minus 119895119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(43)

In order to find a 119902-analogue of the formula [16]

(L119899) [119891 (119905) 119892 (119905)] = (L

119899) [119891 (119905)] (L

119899) [119892 (119905)] (44)

we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]

= (L119899119902) [119891 (119905 119902)] sdot

120598(L119899119902) [119892 (119905 119902)]

(45)

where in the matrix multiplication for every term whichincludes D119896

119902119891 we operate with 120598119896 on 119892 We denote this by sdot

120598

This operator can also be iterated compare with [1]

Theorem 15 (a 119902-analogue of [9 page 67]) Consider

P119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)

where the 119902-Pascal matrix is defined in (8)The mappingsP

119899119902[sdot] andW

119899119902[sdot] are linear that is

P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P

119899119902[119891 (119905)] + 119887P

119899119902[119892 (119905)]

W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W

119899119902[119891 (119905)] + 119887W

119899119902[119892 (119905)]

(47)

One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]

P119899119902[119891 (119905 119902)] sdot

120598P119899119902[119892 (119905 119902)] = P

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(48)

P119899119902[119891 (119905 119902)] sdot

120598W119899119902[119892 (119905 119902)] = W

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(49)

The scaling properties can be stated as follows

W119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

(50)

P119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

times diag [1 119886minus1 1198861minus119899]

(51)

Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule

Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as

P119899119902[119891 (119905 119902)]

minus1119902

equiv P119899119902[119891(119905 119902)

minus1

] (52)

Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let

(G119899119902) [119909] (119894 119895)

equiv

119892119894minus119895(119909) (

119894

119895)

119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(53)

Put

119891 (119905 119909 119902) equiv

119899

sum

119896=0

119892119896(119909)

119905119896

119896119902 (54)

a truncated 119902-exponential generating function of 119902-binomialtype We find that

P119899119902[119891 (119905 119909 119902)]

1003816100381610038161003816119905=0 = G119899119902[119909] (55)

By a routine computation we obtain

G119899119902[119909]G119899119902[119910] = G

119899119902[119909 oplus119902119910] (56)

The function argument 119909oplus119902119910 is induced from the corre-

sponding 119902-additionIn the same way we obtain

G119899119902[119909]G1198991119902

[119910] = G119899119902[119909 ⊞119902119910] (57)

By the last formula we can define the inverse ofG119899119902[119909] as

G119899119902[119909]minus1

equiv G1198991119902

[minus119909] (58)

We now return permanently to [9]

Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-

Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ

119902(119909) for 119891(119905) is given by

Φ119902(119909) (119894) equiv Φ

119894119902(119909) 119894 = 0 1 119899 minus 1 (59)

Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899

(D119902119909) 119910 (119909) = P

119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin

(60)

with initial value 119910(0) = 1199100 has the following solution

119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]

10038161003816100381610038161003816119905=01199100 (61)

Proof Since E119902(119860119905)|119905=0

is equal to the unit matrix the solu-tion of (60) is

Φ119902(119909) = E

119902(P119899119902[119891 (119905)]

1003816100381610038161003816119905=0119909) 1199100

= (

infin

sum

119896=0

P119896

119899119902[119891 (119905)])

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

119909119896

119896119902

(62)

6 Journal of Discrete Mathematics

By (47) and (48)

Φ119902(119909) = P

119899119902(

infin

sum

119896=0

119891119896

(119905)119909119896

119896119902)

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

(63)

where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot

120598

Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if

Φ (119909) = P119899119902[119891 (119905)]

1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(64)

Proof We know from [2 17] thatP119899119902[119905]|119905=0

= 119867119899119902 (65)

We thus obtainΦ (119909) = P

119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]

1003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(66)

Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by

P119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0 (67)

Can be compared with (53)

Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)

Ξ120584119902(119909) andΨ

120584119902(119909) be the 119902-Appell polynomials corresponding

to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then

Ξ120584119902(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119896119902(119911) (68)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)] sdot

120598W119899119902[ℎ (119905)E

119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 oplus119902119911) 119905)]

10038161003816100381610038161003816119905=0

(69)

By (11) the left-hand side of (69) is equal to

(((((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))))

)

times(

Ψ0119902(119911)

Ψ1119902(119911)

Ψ119899minus1119902

(119911)

)

(70)

By (11) the right-hand side of (69) is equal to

(

Ξ0119902(119910 oplus119902119911)

Ξ1119902(119910 oplus119902119911)

Ξ119899minus1119902

(119910 oplus119902119911)

) (71)

The theorem follows by equating the last rows of formulas (71)and (70)

Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which

enables inverses

Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902

and Ξ120584119902

be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902

120584be the pseudo 119902-Appell polynomial

for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then

Ξ120584119902(119910 ⊞119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119902

119896(119911) (72)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)]

10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E

1119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 ⊞119902119911) 119905)] |

119905=0

(73)

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 3

We have similarly

D119902Φ119902

120584(119909) = 120584

119902Φ119902

120584minus1119902(119902119909)

Φ119902

120584(119909) = (Φ

119902⊞119902119909)120584

(15)

22 119902-Analogue of Carlitzrsquo Product Formula (1963) To moti-vate the generalized 119902-Pascal functional matrix we show thatthe Carlitz [3 page 247] formula for the product of generatingfunctions for Appell polynomials has a 119902-analogue involvingthe NWA 119902-addition Define the numbers Φ1015840

120584119902infin

120584=0by the

inverse of (12) (this is allowable since 119891(0) = 1)

119891(119905)minus1

=

infin

sum

120584=0

119905120584

120584119902Φ1015840

120584119902 Φ1015840

0119902= 1 (16)

Then we have the following

Theorem 5 An inverse relation formula for the 119902-Appellpolynomials

119909119899

=

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902Φ119896119902(119909) (17)

Proof By the umbral definition of 119902-Appell polynomials wehave

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902Φ119896119902(119909)

=

119899

sum

119896=0

(119899

119896)

119902

Φ1015840

119899minus119896119902

119896

sum

119897=0

(119896

119897)

119902

Φ119896minus119897119902

119909119897

=

119899

sum

119896=0

119899119902Φ1015840

119899minus119896119902

119899 minus 119896119902

119899

sum

119897=0

Φ119896minus119897119902

119897119902119896 minus 119897

119902119909119897

=

119899

sum

119897=0

119899119902

119897119902119909119897

sum

119904+119905=119899minus119897

Φ1015840

119904119902Φ119905119902

119904119902119905119902

=

119899

sum

119897=0

119899119902

119897119902119909119897

120575119899minus119897

= 119909119899

(18)

This completes the proof

We introduce two other 119902-Appell polynomials Ξ119902and Ψ

119902

by

119892 (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Ξ120584119902(119909) 119892 (119905) isin C [[119909]]

119892 (119905) =

infin

sum

120584=0

119905120584

120584119902Ξ120584119902 Ξ0119902

= 1

ℎ (119905)E119902(119909119905) =

infin

sum

120584=0

119905120584

120584119902Ψ120584119902(119909) ℎ (119905) isin C [[119909]]

ℎ (119905) =

infin

sum

120584=0

119905120584

120584119902Ψ120584119902 Ψ0119902

= 1

(19)

Let us consider119891119898(119886119909) 119892

119899(119887119909)

=

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

119909119903+119904

(20)

Now put

ℎ(119905)minus1

=

infin

sum

120584=0

119905120584

120584119902Ψ1015840

120584119902 (21)

Because of (17) we get

119891119898(119886119909) 119892

119899(119887119909)

=

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

119903+119904

sum

119895=0

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902Ψ119895119902(119909)

(22)

Introduce the notation119860 (119895 119902119898 119899 119886 119887)

equiv

119898

sum

119903=0

119899

sum

119904=0

(119898

119903)

119902

(119899

119904)

119902

(119903 + 119904

119895)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

Ψ1015840

119903+119904minus119895119902

(23)

where

119891119898(119886119909) 119892

119899(119887119909) =

119898+119899

sum

119895=0

119860 (119895 119902119898 119899 119886 119887)Ψ119895119902(119909) (24)

Theorem 6 With the previously mentioned notation one hasthe following 119902-analogue of Carlitz [3 page 247]

infin

sum

119898=0

infin

sum

119899=0

119860 (119895 119902119898 119899 119886 119887)119906119898

119907119899

119898119902119899119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=0

Ψ1015840

119896119902

(119886119906 oplus119902119887119907)119896+119895

119895119902119896119902

(25)

Proof Thus we haveinfin

sum

119898=0

infin

sum

119899=0

119860 (119895 119902119898 119899 119886 119887)119906119898

119907119899

119898119902119899119902

=

infin

sum

119898=0

infin

sum

119899=0

119906119898

119907119899

119898119902119899119902

119898

sum

119903=0

119899

sum

119904=0

= (119898

119903)

119902

(119899

119904)

119902

(119903 + 119904

119895)

119902

119891119898minus119903

119892119899minus119904119886119903

119887119904

Ψ1015840

119903+119904minus119895119902

=

infin

sum

119903=0

infin

sum

119904=0

(119886119906)119903

(119887119907)119904

119903119902119904119902

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

infin

sum

119898=0

infin

sum

119899=0

119891119898119892119899

119906119898

119907119899

119898119902119899119902

= 119891 (119906) 119892 (119907)

infin

sum

119903=0

infin

sum

119904=0

(119886119906)119903

(119887119907)119904

119903119902119904119902

4 Journal of Discrete Mathematics

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902sum

119903+119904=119896

(119886119906)119903

(119887119907)119904

119903119902119904119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902

(119886119906 oplus119902119887119907)119896

119896119902

(26)

23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]

Definition 7 We denote by A119902the set of all 119902-Appell-

numbersLet Φ

119899119902and Ψ

119899119902be two elements in A

119902 Then the

operations + and ⋆ are defined as follows

(Φ119902+ Ψ119902)119899

equiv Φ119899119902+ Ψ119899119902 (27)

(Φ119902⋆ Ψ119902)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

Φ119899minus119896119902

Ψ119896119902 (28)

Theorem 8 The identity element in A119902with respect to ⋆ is

119868 equiv 1205751198990 This corresponds to 119891(119905) = 1

Proof Put 119896 = 119899 and Φ119902= 1 in (28)

Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902

andΨ119899119902

be three elements in A119902corresponding to the generating

functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then

Φ119899119902+ Ψ119899119902

isin A119902

Φ119899119902+ Ψ119899119902

has generating function119891 (119905) + ℎ (119905) (29)

The associative law for addition reads

Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ

119899119902+ Ξ119899119902) + Ψ119899119902 (30)

Proof This follows from the definitions

Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and

Ψ119899119902

be three elements in A119902corresponding to the generating

functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then

Φ119899119902⋆ Ψ119899119902

isin A119902 (31)

Φ119899119902⋆ Ψ119899119902

= Ψ119899119902⋆ Φ119899119902 (32)

Φ119899119902⋆ Ψ119899119902

has generating function119891 (119905) ℎ (119905) (33)

The associative law for ⋆ reads

Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ

119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)

Proof We prove (31) and (33) at the same time We have inthe umbral sense

119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)

Formulas (32) and (34) follow by the commutativity andthe associativity of NWA

24 Specializations

Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by

1199051

(E119902(119905) minus 1)

E119902(119909119905) =

infin

sum

120584=0

119905120584BNWA120584119902 (119909)

120584119902

|119905| lt 2120587 (36)

The NWA 119902-Bernoulli polynomials are also given by

BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584

(37)

The pseudo-119902-Bernoulli polynomials are defined by

B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584

(38)

The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by

1199051

(E1119902

(119905) minus 1)E119902(119909119905) =

infin

sum

120584=0

119905120584BJHC120584119902 (119909)

120584119902

|119905| lt 2120587 (39)

3 119902-Pascal and 119902-Wronskian Matrices

We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread

Definition 12 The generalized 119902-Pascal functional matrix isgiven by

(P119899119902) [119891 (119905 119902)] (119894 119895)

equiv

(119894

119895)

119902

D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(40)

In the special case 119891(119905 119902) = constant

(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)

Definition 13 The 119902-deformed Wronskian vector is given by

(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894

119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1

(42)

Journal of Discrete Mathematics 5

Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)

equiv

D119894minus119895119902119905119891 (119905 119902)

119894 minus 119895119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(43)

In order to find a 119902-analogue of the formula [16]

(L119899) [119891 (119905) 119892 (119905)] = (L

119899) [119891 (119905)] (L

119899) [119892 (119905)] (44)

we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]

= (L119899119902) [119891 (119905 119902)] sdot

120598(L119899119902) [119892 (119905 119902)]

(45)

where in the matrix multiplication for every term whichincludes D119896

119902119891 we operate with 120598119896 on 119892 We denote this by sdot

120598

This operator can also be iterated compare with [1]

Theorem 15 (a 119902-analogue of [9 page 67]) Consider

P119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)

where the 119902-Pascal matrix is defined in (8)The mappingsP

119899119902[sdot] andW

119899119902[sdot] are linear that is

P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P

119899119902[119891 (119905)] + 119887P

119899119902[119892 (119905)]

W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W

119899119902[119891 (119905)] + 119887W

119899119902[119892 (119905)]

(47)

One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]

P119899119902[119891 (119905 119902)] sdot

120598P119899119902[119892 (119905 119902)] = P

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(48)

P119899119902[119891 (119905 119902)] sdot

120598W119899119902[119892 (119905 119902)] = W

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(49)

The scaling properties can be stated as follows

W119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

(50)

P119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

times diag [1 119886minus1 1198861minus119899]

(51)

Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule

Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as

P119899119902[119891 (119905 119902)]

minus1119902

equiv P119899119902[119891(119905 119902)

minus1

] (52)

Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let

(G119899119902) [119909] (119894 119895)

equiv

119892119894minus119895(119909) (

119894

119895)

119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(53)

Put

119891 (119905 119909 119902) equiv

119899

sum

119896=0

119892119896(119909)

119905119896

119896119902 (54)

a truncated 119902-exponential generating function of 119902-binomialtype We find that

P119899119902[119891 (119905 119909 119902)]

1003816100381610038161003816119905=0 = G119899119902[119909] (55)

By a routine computation we obtain

G119899119902[119909]G119899119902[119910] = G

119899119902[119909 oplus119902119910] (56)

The function argument 119909oplus119902119910 is induced from the corre-

sponding 119902-additionIn the same way we obtain

G119899119902[119909]G1198991119902

[119910] = G119899119902[119909 ⊞119902119910] (57)

By the last formula we can define the inverse ofG119899119902[119909] as

G119899119902[119909]minus1

equiv G1198991119902

[minus119909] (58)

We now return permanently to [9]

Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-

Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ

119902(119909) for 119891(119905) is given by

Φ119902(119909) (119894) equiv Φ

119894119902(119909) 119894 = 0 1 119899 minus 1 (59)

Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899

(D119902119909) 119910 (119909) = P

119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin

(60)

with initial value 119910(0) = 1199100 has the following solution

119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]

10038161003816100381610038161003816119905=01199100 (61)

Proof Since E119902(119860119905)|119905=0

is equal to the unit matrix the solu-tion of (60) is

Φ119902(119909) = E

119902(P119899119902[119891 (119905)]

1003816100381610038161003816119905=0119909) 1199100

= (

infin

sum

119896=0

P119896

119899119902[119891 (119905)])

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

119909119896

119896119902

(62)

6 Journal of Discrete Mathematics

By (47) and (48)

Φ119902(119909) = P

119899119902(

infin

sum

119896=0

119891119896

(119905)119909119896

119896119902)

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

(63)

where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot

120598

Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if

Φ (119909) = P119899119902[119891 (119905)]

1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(64)

Proof We know from [2 17] thatP119899119902[119905]|119905=0

= 119867119899119902 (65)

We thus obtainΦ (119909) = P

119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]

1003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(66)

Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by

P119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0 (67)

Can be compared with (53)

Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)

Ξ120584119902(119909) andΨ

120584119902(119909) be the 119902-Appell polynomials corresponding

to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then

Ξ120584119902(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119896119902(119911) (68)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)] sdot

120598W119899119902[ℎ (119905)E

119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 oplus119902119911) 119905)]

10038161003816100381610038161003816119905=0

(69)

By (11) the left-hand side of (69) is equal to

(((((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))))

)

times(

Ψ0119902(119911)

Ψ1119902(119911)

Ψ119899minus1119902

(119911)

)

(70)

By (11) the right-hand side of (69) is equal to

(

Ξ0119902(119910 oplus119902119911)

Ξ1119902(119910 oplus119902119911)

Ξ119899minus1119902

(119910 oplus119902119911)

) (71)

The theorem follows by equating the last rows of formulas (71)and (70)

Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which

enables inverses

Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902

and Ξ120584119902

be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902

120584be the pseudo 119902-Appell polynomial

for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then

Ξ120584119902(119910 ⊞119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119902

119896(119911) (72)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)]

10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E

1119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 ⊞119902119911) 119905)] |

119905=0

(73)

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Discrete Mathematics

(119903 + 119904

119895)

119902

Ψ1015840

119903+119904minus119895119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902sum

119903+119904=119896

(119886119906)119903

(119887119907)119904

119903119902119904119902

= 119891 (119906) 119892 (119907)

infin

sum

119896=119895

(119896

119895)

119902

Ψ1015840

119896minus119895119902

(119886119906 oplus119902119887119907)119896

119896119902

(26)

23 Characterization of 119902-Appell Numbers We will now pre-sent a characterization of 119902-Appell numbers comparewithAl-Salam [6]

Definition 7 We denote by A119902the set of all 119902-Appell-

numbersLet Φ

119899119902and Ψ

119899119902be two elements in A

119902 Then the

operations + and ⋆ are defined as follows

(Φ119902+ Ψ119902)119899

equiv Φ119899119902+ Ψ119899119902 (27)

(Φ119902⋆ Ψ119902)119899

equiv

119899

sum

119896=0

(119899

119896)

119902

Φ119899minus119896119902

Ψ119896119902 (28)

Theorem 8 The identity element in A119902with respect to ⋆ is

119868 equiv 1205751198990 This corresponds to 119891(119905) = 1

Proof Put 119896 = 119899 and Φ119902= 1 in (28)

Theorem 9 (Compare with [6 page 34]) Let Φ119899119902 Ξ119899119902

andΨ119899119902

be three elements in A119902corresponding to the generating

functions119891(119905) 119892(119905) and ℎ(119905) respectively Further assume that119891(0) + ℎ(0) = 0 Then

Φ119899119902+ Ψ119899119902

isin A119902

Φ119899119902+ Ψ119899119902

has generating function119891 (119905) + ℎ (119905) (29)

The associative law for addition reads

Φ119899119902+ (Ξ119899119902+ Ψ119899119902) = (Φ

119899119902+ Ξ119899119902) + Ψ119899119902 (30)

Proof This follows from the definitions

Theorem 10 (Compare with [6 page 34]) LetΦ119899119902 Ξ119899119902 and

Ψ119899119902

be three elements in A119902corresponding to the generating

functions 119891(119905) 119892(119905) and ℎ(119905) respectively Then

Φ119899119902⋆ Ψ119899119902

isin A119902 (31)

Φ119899119902⋆ Ψ119899119902

= Ψ119899119902⋆ Φ119899119902 (32)

Φ119899119902⋆ Ψ119899119902

has generating function119891 (119905) ℎ (119905) (33)

The associative law for ⋆ reads

Φ119899119902⋆ (Ξ119899119902⋆ Ψ119899119902) = (Φ

119899119902⋆ Ξ119899119902) ⋆ Ψ119899119902 (34)

Proof We prove (31) and (33) at the same time We have inthe umbral sense

119891 (119905) ℎ (119905) = E119902(119905 (Φ119902oplus119902Ψ119902)) (35)

Formulas (32) and (34) follow by the commutativity andthe associativity of NWA

24 Specializations

Definition 11 The NWA 119902-Bernoulli polynomialsBNWA120584119902(119909) of degree 120584 are defined by

1199051

(E119902(119905) minus 1)

E119902(119909119905) =

infin

sum

120584=0

119905120584BNWA120584119902 (119909)

120584119902

|119905| lt 2120587 (36)

The NWA 119902-Bernoulli polynomials are also given by

BNWA120584119902 (119909) = (BNWA119902 oplus119902 119909)120584

(37)

The pseudo-119902-Bernoulli polynomials are defined by

B119902NWA120584 (119909) = (BNWA119902 ⊞119902 119909)120584

(38)

The JHC 119902-Bernoulli polynomials BJHC120584119902(119909) of degree 120584are defined by

1199051

(E1119902

(119905) minus 1)E119902(119909119905) =

infin

sum

120584=0

119905120584BJHC120584119902 (119909)

120584119902

|119905| lt 2120587 (39)

3 119902-Pascal and 119902-Wronskian Matrices

We now come to the most important part of the paper whichis of interest even for the case 119902 = 1 since the original paperby Yang and Youn is not very widespread

Definition 12 The generalized 119902-Pascal functional matrix isgiven by

(P119899119902) [119891 (119905 119902)] (119894 119895)

equiv

(119894

119895)

119902

D119894minus119895119902119905119891 (119905 119902) if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(40)

In the special case 119891(119905 119902) = constant

(P119899119902) [119891 (119905 119902)] = 119891 (119905 119902) 119868 (41)

Definition 13 The 119902-deformed Wronskian vector is given by

(W119899119902) [119891 (119905 119902)] (119894 0) equiv D119894

119902119905119891 (119905 119902) 119894 = 0 1 119899 minus 1

(42)

Journal of Discrete Mathematics 5

Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)

equiv

D119894minus119895119902119905119891 (119905 119902)

119894 minus 119895119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(43)

In order to find a 119902-analogue of the formula [16]

(L119899) [119891 (119905) 119892 (119905)] = (L

119899) [119891 (119905)] (L

119899) [119892 (119905)] (44)

we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]

= (L119899119902) [119891 (119905 119902)] sdot

120598(L119899119902) [119892 (119905 119902)]

(45)

where in the matrix multiplication for every term whichincludes D119896

119902119891 we operate with 120598119896 on 119892 We denote this by sdot

120598

This operator can also be iterated compare with [1]

Theorem 15 (a 119902-analogue of [9 page 67]) Consider

P119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)

where the 119902-Pascal matrix is defined in (8)The mappingsP

119899119902[sdot] andW

119899119902[sdot] are linear that is

P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P

119899119902[119891 (119905)] + 119887P

119899119902[119892 (119905)]

W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W

119899119902[119891 (119905)] + 119887W

119899119902[119892 (119905)]

(47)

One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]

P119899119902[119891 (119905 119902)] sdot

120598P119899119902[119892 (119905 119902)] = P

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(48)

P119899119902[119891 (119905 119902)] sdot

120598W119899119902[119892 (119905 119902)] = W

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(49)

The scaling properties can be stated as follows

W119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

(50)

P119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

times diag [1 119886minus1 1198861minus119899]

(51)

Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule

Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as

P119899119902[119891 (119905 119902)]

minus1119902

equiv P119899119902[119891(119905 119902)

minus1

] (52)

Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let

(G119899119902) [119909] (119894 119895)

equiv

119892119894minus119895(119909) (

119894

119895)

119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(53)

Put

119891 (119905 119909 119902) equiv

119899

sum

119896=0

119892119896(119909)

119905119896

119896119902 (54)

a truncated 119902-exponential generating function of 119902-binomialtype We find that

P119899119902[119891 (119905 119909 119902)]

1003816100381610038161003816119905=0 = G119899119902[119909] (55)

By a routine computation we obtain

G119899119902[119909]G119899119902[119910] = G

119899119902[119909 oplus119902119910] (56)

The function argument 119909oplus119902119910 is induced from the corre-

sponding 119902-additionIn the same way we obtain

G119899119902[119909]G1198991119902

[119910] = G119899119902[119909 ⊞119902119910] (57)

By the last formula we can define the inverse ofG119899119902[119909] as

G119899119902[119909]minus1

equiv G1198991119902

[minus119909] (58)

We now return permanently to [9]

Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-

Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ

119902(119909) for 119891(119905) is given by

Φ119902(119909) (119894) equiv Φ

119894119902(119909) 119894 = 0 1 119899 minus 1 (59)

Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899

(D119902119909) 119910 (119909) = P

119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin

(60)

with initial value 119910(0) = 1199100 has the following solution

119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]

10038161003816100381610038161003816119905=01199100 (61)

Proof Since E119902(119860119905)|119905=0

is equal to the unit matrix the solu-tion of (60) is

Φ119902(119909) = E

119902(P119899119902[119891 (119905)]

1003816100381610038161003816119905=0119909) 1199100

= (

infin

sum

119896=0

P119896

119899119902[119891 (119905)])

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

119909119896

119896119902

(62)

6 Journal of Discrete Mathematics

By (47) and (48)

Φ119902(119909) = P

119899119902(

infin

sum

119896=0

119891119896

(119905)119909119896

119896119902)

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

(63)

where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot

120598

Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if

Φ (119909) = P119899119902[119891 (119905)]

1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(64)

Proof We know from [2 17] thatP119899119902[119905]|119905=0

= 119867119899119902 (65)

We thus obtainΦ (119909) = P

119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]

1003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(66)

Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by

P119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0 (67)

Can be compared with (53)

Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)

Ξ120584119902(119909) andΨ

120584119902(119909) be the 119902-Appell polynomials corresponding

to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then

Ξ120584119902(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119896119902(119911) (68)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)] sdot

120598W119899119902[ℎ (119905)E

119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 oplus119902119911) 119905)]

10038161003816100381610038161003816119905=0

(69)

By (11) the left-hand side of (69) is equal to

(((((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))))

)

times(

Ψ0119902(119911)

Ψ1119902(119911)

Ψ119899minus1119902

(119911)

)

(70)

By (11) the right-hand side of (69) is equal to

(

Ξ0119902(119910 oplus119902119911)

Ξ1119902(119910 oplus119902119911)

Ξ119899minus1119902

(119910 oplus119902119911)

) (71)

The theorem follows by equating the last rows of formulas (71)and (70)

Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which

enables inverses

Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902

and Ξ120584119902

be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902

120584be the pseudo 119902-Appell polynomial

for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then

Ξ120584119902(119910 ⊞119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119902

119896(119911) (72)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)]

10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E

1119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 ⊞119902119911) 119905)] |

119905=0

(73)

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 5

Definition 14 The 119902-deformed Leibniz functional matrix isgiven by(L119899119902) [119891 (119905 119902)] (119894 119895)

equiv

D119894minus119895119902119905119891 (119905 119902)

119894 minus 119895119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(43)

In order to find a 119902-analogue of the formula [16]

(L119899) [119891 (119905) 119892 (119905)] = (L

119899) [119891 (119905)] (L

119899) [119892 (119905)] (44)

we infer that by the 119902-Leibniz formula [1](L119899119902) [119891 (119905 119902) 119892 (119905 119902)]

= (L119899119902) [119891 (119905 119902)] sdot

120598(L119899119902) [119892 (119905 119902)]

(45)

where in the matrix multiplication for every term whichincludes D119896

119902119891 we operate with 120598119896 on 119892 We denote this by sdot

120598

This operator can also be iterated compare with [1]

Theorem 15 (a 119902-analogue of [9 page 67]) Consider

P119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0= 119875119899119902(119909) (46)

where the 119902-Pascal matrix is defined in (8)The mappingsP

119899119902[sdot] andW

119899119902[sdot] are linear that is

P119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886P

119899119902[119891 (119905)] + 119887P

119899119902[119892 (119905)]

W119899119902[119886119891 (119905) + 119887119892 (119905)] = 119886W

119899119902[119891 (119905)] + 119887W

119899119902[119892 (119905)]

(47)

One has the following two matrix multiplication formulasthe first one is a 119902-analogue of [8 page 232]

P119899119902[119891 (119905 119902)] sdot

120598P119899119902[119892 (119905 119902)] = P

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(48)

P119899119902[119891 (119905 119902)] sdot

120598W119899119902[119892 (119905 119902)] = W

119899119902[119891 (119905 119902) 119892 (119905 119902)]

(49)

The scaling properties can be stated as follows

W119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]W119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

(50)

P119899119902[119891 (119886119905 119902)]

1003816100381610038161003816119905=0

= diag [1 119886 119886119899minus1]P119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0

times diag [1 119886minus1 1198861minus119899]

(51)

Proof The formula (49) is proved by the 119902-Leibniz theoremFor (50) use the chain rule

Definition 16 Assume that 119891(0) = 0 If the inverse (119891(119905)minus1)(119896)exists for 119896 lt 119899 we can define the 119902-inverse of the generalized119902-Pascal functional matrix as

P119899119902[119891 (119905 119902)]

minus1119902

equiv P119899119902[119891(119905 119902)

minus1

] (52)

Example 17 (a 119902-analogue of [8 pages 231ndash234]) Let

(G119899119902) [119909] (119894 119895)

equiv

119892119894minus119895(119909) (

119894

119895)

119902

if 119894 ge 119895

119894 119895 = 0 1 2 119899 minus 1

0 otherwise(53)

Put

119891 (119905 119909 119902) equiv

119899

sum

119896=0

119892119896(119909)

119905119896

119896119902 (54)

a truncated 119902-exponential generating function of 119902-binomialtype We find that

P119899119902[119891 (119905 119909 119902)]

1003816100381610038161003816119905=0 = G119899119902[119909] (55)

By a routine computation we obtain

G119899119902[119909]G119899119902[119910] = G

119899119902[119909 oplus119902119910] (56)

The function argument 119909oplus119902119910 is induced from the corre-

sponding 119902-additionIn the same way we obtain

G119899119902[119909]G1198991119902

[119910] = G119899119902[119909 ⊞119902119910] (57)

By the last formula we can define the inverse ofG119899119902[119909] as

G119899119902[119909]minus1

equiv G1198991119902

[minus119909] (58)

We now return permanently to [9]

Definition 18 (compare with [9 page 71]) LetΦ119902(119909) be the 119902-

Appell polynomial corresponding to 119891(119905) in (11) The vectorof 119902-Appell polynomials Φ

119902(119909) for 119891(119905) is given by

Φ119902(119909) (119894) equiv Φ

119894119902(119909) 119894 = 0 1 119899 minus 1 (59)

Theorem 19 (a 119902-analogue of [9 page 69]) Let 119891(119905) beanalytic in the neighbourhood of 0 with 119891(0) = 0 The 119902-difference equation in R119899

(D119902119909) 119910 (119909) = P

119899119902[119891 (119905 119902)]

1003816100381610038161003816119905=0 119910 (119909) minusinfin lt 119909 lt infin

(60)

with initial value 119910(0) = 1199100 has the following solution

119910 (119909) = P119899119902[E119902(119891 (119905) 119909)]

10038161003816100381610038161003816119905=01199100 (61)

Proof Since E119902(119860119905)|119905=0

is equal to the unit matrix the solu-tion of (60) is

Φ119902(119909) = E

119902(P119899119902[119891 (119905)]

1003816100381610038161003816119905=0119909) 1199100

= (

infin

sum

119896=0

P119896

119899119902[119891 (119905)])

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

119909119896

119896119902

(62)

6 Journal of Discrete Mathematics

By (47) and (48)

Φ119902(119909) = P

119899119902(

infin

sum

119896=0

119891119896

(119905)119909119896

119896119902)

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

(63)

where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot

120598

Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if

Φ (119909) = P119899119902[119891 (119905)]

1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(64)

Proof We know from [2 17] thatP119899119902[119905]|119905=0

= 119867119899119902 (65)

We thus obtainΦ (119909) = P

119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]

1003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(66)

Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by

P119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0 (67)

Can be compared with (53)

Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)

Ξ120584119902(119909) andΨ

120584119902(119909) be the 119902-Appell polynomials corresponding

to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then

Ξ120584119902(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119896119902(119911) (68)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)] sdot

120598W119899119902[ℎ (119905)E

119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 oplus119902119911) 119905)]

10038161003816100381610038161003816119905=0

(69)

By (11) the left-hand side of (69) is equal to

(((((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))))

)

times(

Ψ0119902(119911)

Ψ1119902(119911)

Ψ119899minus1119902

(119911)

)

(70)

By (11) the right-hand side of (69) is equal to

(

Ξ0119902(119910 oplus119902119911)

Ξ1119902(119910 oplus119902119911)

Ξ119899minus1119902

(119910 oplus119902119911)

) (71)

The theorem follows by equating the last rows of formulas (71)and (70)

Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which

enables inverses

Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902

and Ξ120584119902

be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902

120584be the pseudo 119902-Appell polynomial

for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then

Ξ120584119902(119910 ⊞119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119902

119896(119911) (72)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)]

10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E

1119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 ⊞119902119911) 119905)] |

119905=0

(73)

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Discrete Mathematics

By (47) and (48)

Φ119902(119909) = P

119899119902(

infin

sum

119896=0

119891119896

(119905)119909119896

119896119902)

1003816100381610038161003816100381610038161003816100381610038161003816119905=0

(63)

where the power 119891119896(119905) is to be interpreted as 119896 minus 1 productswith sdot

120598

Theorem 20 (a 119902-analogue of [9 page 71]) Let Φ(119909) be avector of length 119899 Then Φ(119909) is the 119902-Appell polynomial vectorfor 119891(119905) if and only if

Φ (119909) = P119899119902[119891 (119905)]

1003816100381610038161003816119905=0sdot120598W119899119902 [E119902 (119909119905)]10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(64)

Proof We know from [2 17] thatP119899119902[119905]|119905=0

= 119867119899119902 (65)

We thus obtainΦ (119909) = P

119899119902[E119902(119909119905)]

10038161003816100381610038161003816119905=0sdot120598W119899119902[119891 (119905)]

1003816100381610038161003816119905=0

= W119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(66)

Definition 21 (A 119902-analogue of [9 page 74]) Let Φ(119909) be the119902-Appell polynomial vector for119891(119905)Then the 119902-Pascalmatrixfor the 119902-Appell polynomial vector for 119891(119905) is defined by

P119899119902[119891 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0 (67)

Can be compared with (53)

Theorem 22 (a 119902-analogue of [9 pages 74-75]) Let Φ120584119902(119909)

Ξ120584119902(119909) andΨ

120584119902(119909) be the 119902-Appell polynomials corresponding

to 119891(119905) 119892(119905) and ℎ(119905) respectivelyFurther assume that 119891(119905)ℎ(119905) = 119892(119905)Then

Ξ120584119902(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119896119902(119911) (68)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)] sdot

120598W119899119902[ℎ (119905)E

119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 oplus119902119911) 119905)]

10038161003816100381610038161003816119905=0

(69)

By (11) the left-hand side of (69) is equal to

(((((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))))

)

times(

Ψ0119902(119911)

Ψ1119902(119911)

Ψ119899minus1119902

(119911)

)

(70)

By (11) the right-hand side of (69) is equal to

(

Ξ0119902(119910 oplus119902119911)

Ξ1119902(119910 oplus119902119911)

Ξ119899minus1119902

(119910 oplus119902119911)

) (71)

The theorem follows by equating the last rows of formulas (71)and (70)

Formula (68) is almost a generalization of (28)There is a variation of this formula with JHC which

enables inverses

Theorem 23 (another 119902-analogue of [9 pages 74-75]) LetΦ120584119902

and Ξ120584119902

be two 119902-Appell polynomial sequences for 119891(119905)and119892(119905) respectively LetΨ119902

120584be the pseudo 119902-Appell polynomial

for ℎ(119905) Furthermore assume that 119891(119905)ℎ(119905) = 119892(119905) Then

Ξ120584119902(119910 ⊞119902119911) =

120584

sum

119896=0

(120584

119896)

119902

Φ120584minus119896119902

(119910)Ψ119902

119896(119911) (72)

Proof By formula (49) we get

P119899119902[119891 (119905)E

119902(119910119905)]

10038161003816100381610038161003816119905=0W119899119902[ℎ (119905)E

1119902(119911119905)]

10038161003816100381610038161003816119905=0

= W119899119902[119891 (119905) ℎ (119905)E

119902((119910 ⊞119902119911) 119905)] |

119905=0

(73)

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 7

By (11) and (13) the left-hand side of (73) is equal to

(((

(

Φ0119902(119910) 0 sdot sdot sdot 0

Φ1119902(119910) Φ

0119902(119910) sdot sdot sdot 0

Φ2119902(119910) 2

119902Φ1119902(119910) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119910) (119899 minus 1

1)

119902

Φ119899minus2119902

(119910) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119910)

)))

)

times(

Ψ119902

0(119911)

Ψ119902

1(119911)

Ψ119902

119899minus1(119911)

)

(74)

By (11) the right-hand side of (73) is equal to

(

Ξ0119902(119910 ⊞119902119911)

Ξ1119902(119910 ⊞119902119911)

Ξ119899minus1119902

(119910 ⊞119902119911)

) (75)

The theorem follows by equating the last rows of formulas(75) and (74)

The following formula from [18 equation (50) page 133]for 119902 = 1 follows immediately from (68)

Corollary 24 ([11 4242]) When sum119904119897=1

119899119897= 119899

B(119899)119873119882119860119896119902

(1199091oplus119902sdot sdot sdot oplus119902119909119904)

= sum

1198981+sdotsdotsdot+119898

119904=119896

(119896

1198981 119898

119904

)

119902

119904

prod

119895=1

B(119899119895)119873119882119860119898

119895119902(119909119895)

(76)

where we assume that 119899119895operates on 119909

119895

Yang and Youn have found a new identity betweenBernoulli and Euler polynomials

2119899B119899(119910 + 119911

2)

120584

sum

119896=0

(120584

119896)B119896(119910)E119899minus119896

(119911) (77)

The 119902-analogue looks as follows

Corollary 25 (a 119902-analogue of [9 page 75 (18)]) Consider

L119873119882119860120584119902

(119910 oplus119902119911) =

120584

sum

119896=0

(120584

119896)

119902

B119873119882119860120584minus119896119902

(119910) F119873119882119860120584119902

(119911)

(78)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and ℎ(119905) = 2(E

119902(119905)+1)

in (68) Then

119892119899(119905) = 119891 (119905) 119892 (119905) =

(2119905)1

(E119902(1199052119902) minus 1)

(79)

This is obviously the same as the left hand side of thegenerating function for 119902 L-numbers

Corollary 26 (a 119902-analogue of [9 pages 85-86]) Let Φ119899119902(119909)

be a 119902-Appell polynomial andΨ119902119899(119909) a pseudo 119902-Appell polyno-

mial Then119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(minus119909) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(80)

Let Φ119899119902(119909) and Ψ

119899119902(119909) be 119902-Appell polynomials Then

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910)Ψ119899minus119896119902

(119911) =

119899

sum

119896=0

(119899

119896)

119902

Φ119896119902(119910 oplus119902119911)Ψ119899minus119896119902

(81)

Proof Formula (80) is proved as follows We put 119911 = minus119910 in(72) The RHS of (72) is then equal to the LHS of (80) Wefinish the proof by using formula (49) Formula (81) is provedas follows Use (68) with 119891(119905) rarr 119891(119905)E

119902(119905(119910 oplus

119902119911)) and ℎ(119905)

Or use an equivalent umbral reasoning

Corollary 27 One has the following special cases119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119909)B119902119873119882119860119899minus119896

(minus119909)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

B119873119882119860119899minus119896119902

(82)

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(83)

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910)B119873119882119860119899minus119896119902

(119911)

=

119899

sum

119896=0

(119899

119896)

119902

F119873119882119860119896119902

(119910 oplus119902119911)B119873119882119860119899minus119896119902

(84)

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Discrete Mathematics

Corollary 28 (a 119902-analogue of [9 page 87]) Consider the 119902-Appell polynomialΦ

119899119902(119909) and the pseudo-119902-Appell polynomial

Ψ119902

119899(119909) Then119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902(119909)Ψ119902

119899minus119896(119909) =

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

Φ119896119902Ψ119899minus119896119902

(85)

Proof Let Φ119899119902(119909) be the 119902-Appell polynomial for 119891(119905) and

Ψ119902

119899(119909) the pseudo-119902-Appell polynomial for ℎ(119905) By formula

(50)

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0

=((

(

Φ0119902(119909) 0 sdot sdot sdot 0

Φ1119902(119909) Φ

0119902(119909) sdot sdot sdot 0

Φ2119902(119909) 2

119902Φ1119902(119909) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909) (119899 minus 1

1)

119902

Φ119899minus2119902

(119909) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909)

))

)

(

Ψ119902

0(119909)

minusΨ119902

1(119909)

(minus1)119899+1

Ψ119902

119899minus1(119909)

)

(86)

On the other hand by formula (49) we obtain

P119899119902[119891 (119905)E

119902(119909119905)] sdot

120598W119899119902[ℎ (minus119905)E

1119902(minus119909119905)]

10038161003816100381610038161003816119905=0= P119899119902[119891 (119905)] sdot

120598W119899119902[ℎ (minus119905)]|

119905=0

=((

(

Φ0119902(0) 0 sdot sdot sdot 0

Φ1119902(0) Φ

0119902(0) sdot sdot sdot 0

Φ2119902(0) 2

119902Φ1119902(0) sdot sdot sdot 0

(119899 minus 1

0)

119902

Φ119899minus1119902

(0) (119899 minus 1

1)

119902

Φ119899minus2119902

(0) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(0)

))

)

(

Ψ119902

0(0)

minusΨ119902

1(0)

(minus1)119899+1

Ψ119902

119899minus1(0)

)

(87)

The theorem follows by multiplying both sides of (85) by(minus1)119899 and equating the last rows of formulas (86) and

(87)

We can easily write down specializations of (85) to(pseudo) 119902-Bernoulli and 119902-Euler polynomials One exampleis

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902 (119909)B119902

NWA119899minus119896 (119909)

=

119899

sum

119896=0

(minus1)119896

(119899

119896)

119902

BNWA119896119902BNWA119899minus119896119902

(88)

Theorem 29 (the 119902-connection constant theorem a 119902-anal-ogue of [9 page 77]) Let Φ

119902(119909) and Ψ

119902(119909) be the 119902-Appell

vectors corresponding to 119891(119905) and 119892(119905) respectively Then

Φ119902(119909) = P

119899119902[119891 (119905)

119892 (119905)]

10038161003816100381610038161003816100381610038161003816119905=0

Ψ119902(119909) (89)

Proof By formula (49) we get

Φ119902(119909) = W

119899119902[119891 (119905)

119892 (119905)

119892 (119905)E119902(119909119905)]

10038161003816100381610038161003816100381610038161003816119905=0

= P119899119902[119891 (119905)

119892 (119905)] sdot120598W119899119902[119892 (119905)E

119902(119909119905)]

10038161003816100381610038161003816119905=0

(90)

Corollary 30 (a 119902-analogue of [9 page 77]) A relationshipbetween Bernoulli and Euler polynomials

B119873119882119860119899119902

(119909) =119899119902F119873119882119860119899minus1119902

(119909)

2

+

119899

sum

119896=0

(119899

119896)

119902

B119873119882119860119899minus119896119902

F119873119882119860119896119902

(119909)

(91)

Proof We put in119891(119905) = 1199051(E119902(119905)minus1) and 119892(119905) = 2(E

119902(119905)+1)

in (89) Then

119891 (119905)

119892 (119905)=1199051

2+

1199051

E119902(119905) minus 1

(92)

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Discrete Mathematics 9

We find that

(119891 (119905)

119892 (119905))

(119896)

(0) =

1

2+ BNWA119896119902 if 119896 = 1

BNWA119896119902 otherwise

(93)

The result follows by (89)

Theorem 31 (a 119902-analogue of [9 page 91]) LetΦ119899119902(119909) be the

119902-Appell polynomial sequence for 119891(119905) Then Φ119899119902(119909) satisfies

the following 119902-difference equation

119899

sum

119896=2

119902119899

120572119896minus1

119896 minus 1119902D119896119902119891 (119909119902

minus1

)

+ 119902119899

(119909119902 + 1205720)D119902119891 (119909119902

minus1

)

minus 119899 + 1119902119891 (119909) + 119902

119899

119891 (119909119902minus1

) = 0

(94)

where 120572119896equiv D119896119902(minus119891(119905)D

119902(1119891(119905)))|

119905=0

Proof We put 119891(119905) = 1119892(119905)

(

Φ1119902(119909)

Φ2119902(119909)

Φ119899119902(119909)

) = W119899119902[D119902(E119902(119909119905)

119892 (119905))]

100381610038161003816100381610038161003816100381610038161003816119905=0

= W119899119902[(119909 minus

D119902119892 (119905)

119892 (119905))E119902(119909119905)

119892 (119902119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

by (50)= P

119899119902[E119902(119909119905)

119892 (119902119905)] sdot120598W119899119902[119909 minus

D119902119892 (119905)

119892 (119905)]

100381610038161003816100381610038161003816100381610038161003816119905=0

=(((

(

Φ0119902(119909119902minus1

) 0 sdot sdot sdot 0

119902Φ1119902(119909119902minus1

) Φ0119902(119909119902minus1

) sdot sdot sdot 0

1199022

Φ2119902(119909119902minus1

) 1199022119902Φ1119902(119909119902minus1

) sdot sdot sdot 0

119902119899minus1

(119899 minus 1

0)

119902

Φ119899minus1119902

(119909119902minus1

) 119902119899minus2

(119899 minus 1

1)

119902

Φ119899minus2119902

(119909119902minus1

) sdot sdot sdot (119899 minus 1

119899 minus 1)

119902

Φ0119902(119909119902minus1

)

)))

)

(

119909+ 1205720

1205721

120572119899minus1

)

(95)

By equating the first and last expression for the last rows offormula (95) we get (120584 = 0 1 119899)

Φ120584119902(119909) = (119909 + 120572

0) 119902120584minus1

Φ120584minus1119902

(119909119902minus1

)

+

120584minus1

sum

119896=1

(120584 minus 1

119896)

119902

120572119896119902120584minus119896minus1

Φ120584minus119896minus1119902

(119909119902minus1

)

(96)

Operating with D119902on this and putting 120584 minus 1 = 119899

119899 + 1119902Φ119899119902(119909)

= 119902119899

Φ119899119902(119909119902minus1

)

+ 119902119899minus1

119899119902(119902119909 + 120572

0)Φ119899minus1119902

(119909119902minus1

)

+

119899

sum

119896=1

(119899

119896)

119902

120572119896119902119899minus119896minus1

119899 minus 119896119902

times Φ119899minus119896minus1119902

(119909119902minus1

) 119899 = 0 1 119899

(97)

where we have used the following identities

D119896119902Φ119899119902(119909119902minus1

) = 119902minus119896

119899 minus 119896 + 1119896119902Φ119899minus119896119902

D119899119902Φ119899minus1119902

(119909119902minus1

) = 0

(98)

4 Conclusion

By finding 119902-analogues of Yang and Youn [9] we have alsoshown that Yang-Youn paper contains a lot of new and inter-esting formulas It is our hope that this paper together withfuture articles will point out the interplay between specialfunctions and linear algebra The formulas in Corollary 28are in the same style as Noslashrlundrsquos [18] formulas which areproved using umbral calculus However our formulas (78)Corollary 30 and (91) are of a more intricate nature Thisshows that the matrix approach in this paper is not just arestatement of Noslashrlundrsquos [18] umbral calculus in 119902-deformedform but a generalization We also see that in order to obtain119902-analogues of the Yang and Youn [9 pages 85-87] formulaswe had to use (pseudo) 119902-Bernoulli and 119902-Euler polynomialswhich were first mentioned in [5]

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Discrete Mathematics

Appendix

Bernoulli Number Formulas in Hansen

In Hansens impressive table [10 pages 331ndash347] severalformulas for Bernoulli and Euler numbers are listedWe showthat some of these are connected to our formulas For 119902 = 1formula (83) can be written as follows

119899

sum

119896=0

(119899

119896)B119896(119910)B119899minus119896

(119911)

=

119899

sum

119896=0

(119899

119896)B119896(119910 + 119911)B

119899minus119896

(A1)

Hansen [10 page 341 50112] writes this as follows

119899

sum

119896=0

(minus1)119896(minus119899)119896

119896B119896(119910)B119899minus119896

(119911)

= 119899 (119910 + 119911 minus 1)B119899minus1

(119910 + 119911) minus (119899 minus 1)B119899(119910 + 119911)

(A2)

Hansen [10 page 346 5162] also has a corresponding resultfor the Euler polynomials

References

[1] T Ernst ldquo119902-Leibniz functional matrices with connections to 119902-Pascal and 119902-Stirling matricesrdquo Advanced Studies in Contempo-rary Mathematics vol 22 no 4 2012

[2] T Ernst ldquoAn umbral approach for 119902-Pascal matricesrdquo Submit-ted

[3] L Carlitz ldquoProducts of Appell polynomialsrdquo Collectanea Math-ematica vol 15 pp 245ndash258 1963

[4] P Nalli ldquoSopra un procedimento di calcolo analogo alla inte-grazionerdquo Rendiconti del CircoloMatematico di Palermo vol 47no 3 pp 337ndash374 1923

[5] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproach IIrdquo UUDM Report 200917 2009

[6] W A Al-Salam ldquo119902-Appell polynomialsrdquo Annali di MatematicaPura ed Applicata vol 77 no 1 pp 31ndash45 1967

[7] T Ernst ldquo119902-Bernoulli and 119902-Euler polynomials an umbralapproachrdquo International Journal of Difference Equations vol 1no 1 pp 31ndash80 2006

[8] Y Yang and C Micek ldquoGeneralized Pascal functional matrixand its applicationsrdquo Linear Algebra and Its Applications vol423 no 2-3 pp 230ndash245 2007

[9] Y Yang and H Youn ldquoAppell polynomial sequences a linearalgebra approachrdquo JP Journal of Algebra Number Theory andApplications vol 13 no 1 pp 65ndash98 2009

[10] E Hansen A Table of Series and Products Prentice Hall 1975[11] T Ernst A Comprehensive Treatment of q-Calculus Birkhauser

2012[12] W Hahn Lineare geometrische Differenzengleichungen vol 169

Forschungszentrum Graz Mathematisch-Statistische SektionGraz Austria 1981

[13] L M Milne-Thomson The Calculus of Finite DifferencesMacmillan London UK 1951

[14] E D Rainville Special Functions Chelsea Bronx NY USA 1stedition 1971 Reprint of 1960 first edition

[15] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880

[16] Y Yang ldquoGeneralized Leibniz functional matrices and factor-izations of some well-known matricesrdquo Linear Algebra and ItsApplications vol 430 no 1 pp 511ndash531 2009

[17] T Ernst ldquoAn umbral approach to find 119902-analogues of matrixformulasrdquo Submitted

[18] N E Noslashrlund Differenzenrechnung Springer 1924

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of