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Research ArticleOn a Class of š-Bernoulli, š-Euler, and š-Genocchi Polynomials
N. I. Mahmudov and M. Momenzadeh
Eastern Mediterranean University, Gazimagusa, Northern Cyprus, Turkey
Correspondence should be addressed to M. Momenzadeh; [email protected]
Received 20 January 2014; Revised 3 April 2014; Accepted 3 April 2014; Published 28 April 2014
Academic Editor: Sofiya Ostrovska
Copyright Ā© 2014 N. I. Mahmudov and M. Momenzadeh. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
The main purpose of this paper is to introduce and investigate a class of š-Bernoulli, š-Euler, and š-Genocchi polynomials. Theš-analogues of well-known formulas are derived. In addition, the š-analogue of the Srivastava-PinteĢr theorem is obtained. Somenew identities, involving š-polynomials, are proved.
1. Introduction
Throughout this paper, we always make use of the classicaldefinition of quantum concepts as follows.
The š-shifted factorial is defined by
(š; š)0= 1, (š; š)
š=
šā1
ā
š=0
(1 ā ššš) , š ā N,
(š; š)ā=
ā
ā
š=0
(1 ā ššš) ,
š < 1, š ā C.
(1)
It is known that
(š; š)š=
š
ā
š=0
[š
š]
š
š(1/2)š(šā1)
(ā1)ššš. (2)
The š-numbers and š-factorial are defined by
[š]š =1 ā šš
1 ā š, (š Ģø= 1, š ā C) ;
[0]š! = 1, [š]š! = [š]š[š ā 1]š!.
(3)
The š-polynomial coefficient is defined by
[š
š]
š
=(š; š)š
(š; š)šāš(š; š)š
, (š ā©½ š, š, š ā N) . (4)
In the standard approach to the š-calculus two exponen-tial functions are used, these š-exponential functions andimproved type š-exponential function (see [1]) are defined asfollows:
šš(š§) =
ā
ā
š=0
š§š
[š]š!=
ā
ā
š=0
1
(1 ā (1 ā š) ššš§),
0 <š < 1, |š§| <
11 ā š
,
šøš (š§) = š1/š (š§) =
ā
ā
š=0
š(1/2)š(šā1)
š§š
[š]š!
=
ā
ā
š=0
(1 + (1 ā š) ššš§) , 0 <
š < 1, š§ ā C,
Eš(š§) = š
š(š§
2)šøš(š§
2) =
ā
ā
š=0
(ā1, š)š
2š
š§š
[š]š!
=
ā
ā
š=0
(1 + (1 ā š) šš(š§/2))
(1 ā (1 ā š) šš (š§/2)), 0 < š < 1, |š§|<
2
1 ā š.
(5)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 696454, 10 pageshttp://dx.doi.org/10.1155/2014/696454
2 Abstract and Applied Analysis
The form of improved type of š-exponential function Eš(š§)
motivated us to define a new š-addition and š-subtraction as
(š„ āšš¦)š
:=
š
ā
š=0
[š
š]
š
(ā1, š)š(ā1, š)
šāš
2šš„šš¦šāš,
š = 0, 1, 2, . . . ,
(š„āšš¦)š
:=
š
ā
š=0
[š
š]
š
(ā1, š)š(ā1, š)
šāš
2šš„š(āš¦)šāš,
š = 0, 1, 2, . . . .
(6)
It follows that
Eš (š”š„)Eš (š”š¦) =
ā
ā
š=0
(š„ āšš¦)š š”š
[š]š!. (7)
The Bernoulli numbers {šµš}šā„0
are rational numbers in asequence defined by the binomial recursion formula:
š
ā
š=0
(š
š)šµšā šµš= {
1, š = 1,
0, š > 1,(8)
or equivalently, the generating function
ā
ā
š=0
šµš
š”š
š!=
š”
šš” ā 1. (9)
š-Analogues of the Bernoulli numbers were first studiedby Carlitz [2] in the middle of the last century when heintroduced a new sequence {š½
š}šā©¾0
:
š
ā
š=0
(š
š)š½ššš+1
ā š½š= {
1, š = 1,
0, š > 1.(10)
Here and in the remainder of the paper, for the parameter šwe make the assumption that |š| < 1. Clearly we recover (8)if we let š ā 1 in (10). The š-binomial formula is known as
(1 ā š)š
š= (š; š)
š=
šā1
ā
š=0
(1 ā ššš)
=
š
ā
š=0
[š
š]
š
š(1/2)š(šā1)
(ā1)ššš.
(11)
The above š-standard notation can be found in [3].Carlitz has introduced the š-Bernoulli numbers and poly-
nomials in [2]. Srivastava and PinteĢr proved some relationsand theorems between the Bernoulli polynomials and Eulerpolynomials in [4]. They also gave some generalizationsof these polynomials. In [4ā16], the authors investigatedsome properties of the š-Euler polynomials and š-Genocchipolynomials. They gave some recurrence relations. In [17],Cenkci et al. gave the š-extension of Genocchi numbers ina different manner. In [18], Kim gave a new concept for theš-Genocchi numbers and polynomials. In [19], Simsek et al.investigated the š-Genocchi zeta function and š-function by
using generating functions andMellin transformation.Thereare numerous recent studies on this subject by, among manyother authors, Cigler [20], Cenkci et al. [17, 21], Choi et al.[22], Cheon [23], Luo and Srivastava [8ā10], Srivastava et al.[4, 24], Nalci and Pashaev [25] Gaboury and Kurt, [26], Kimet al. [27], and Kurt [28].
We first give the definitions of the š-numbers and š-polynomials. It should be mentioned that the definition of š-Bernoulli numbers in Definition 1 can be found in [25].
Definition 1. Let š ā C, 0 < |š| < 1. The š-Bernoulli numbersbš,š
and polynomials Bš,š(š„, š¦) are defined by means of the
generating functions:
BĢ (š”) :=š”šš(āš”/2)
šš (š”/2) ā šš (āš”/2)
=š”
Eš (š”) ā 1
=
ā
ā
š=0
bš,š
š”š
[š]š!, |š”| < 2š,
š”
Eš (š”) ā 1
Eš(š”š„)E
š(š”š¦)
=
ā
ā
š=0
Bš,š(š„, š¦)
š”š
[š]š!, |š”| < 2š.
(12)
Definition 2. Let š ā C, 0 < |š| < 1. The š-Euler numberseš,š
and polynomials Eš,š(š„, š¦) are defined by means of the
generating functions:
EĢ (š”) :=2šš(āš”/2)
šš (š”/2) + šš (āš”/2)
=2
Eš (š”) + 1
=
ā
ā
š=0
eš,š
š”š
[š]š!, |š”| < š,
2
Eš(š”) + 1
Eš(š”š„)E
š(š”š¦)
=
ā
ā
š=0
Eš,š(š„, š¦)
š”š
[š]š!, |š”| < š.
(13)
Definition 3. Let š ā C, 0 < |š| < 1.The š-Genocchi numbersgš,š
and polynomials Gš,š(š„, š¦) are defined by means of the
generating functions:
GĢ (š”) :=2š”šš(āš”/2)
šš(š”/2) + š
š(āš”/2)
=2š”
Eš(š”) + 1
=
ā
ā
š=0
gš,š
š”š
[š]š!, |š”| < š,
2š”
Eš(š”) + 1
Eš (š”š„)Eš (š”š¦)
=
ā
ā
š=0
Gš,š(š„, š¦)
š”š
[š]š!, |š”| < š.
(14)
Abstract and Applied Analysis 3
Note that Cigler [20] defined š-Genocchi numbers as
š”šš (š”) + šš (āš”)
šš(š”) + š
š(āš”)
=
ā
ā
š=0
š2š,š
(ā1)šā1(āš; š)
2šā1š”2š
[2š]š!. (15)
Then comparing gš,š
with šš,š, we see that
(ā1)šā122š+1
g2š+2,š
= (āš; š)2š+1
š2š+2,š
. (16)
Definition 4. Let š ā C, 0 < |š| < 1. The š-tangent numbersTš,š
are defined by means of the generating functions:
tanhšš” = āš tan
š (šš”) =šš (š”) ā šš (āš”)
šš(š”) + š
š(āš”)
=Eš (2š”) ā 1
Eš(2š”) + 1
=
ā
ā
š=1
T2š+1,š
(ā1)šš”2š+1
[2š + 1]š!.
(17)
It is obvious that, by letting š tend to 1 from the left side,we lead to the classic definition of these polynomials:
bš,š:= Bš,š (0) , lim
šā1ā
Bš,š (š„) = šµš (š„) ,
limšā1
ā
Bš,š(š„, š¦) = šµ
š(š„ + š¦) , lim
šā1ā
bš,š= šµš,
eš,š:= Eš,š (0) , lim
šā1ā
Eš,š (š„) = šøš (š„) ,
limšā1
ā
Eš,š(š„, š¦) = šø
š(š„ + š¦) , lim
šā1ā
eš,š= šøš,
gš,š:= Gš,š(0) , lim
šā1ā
Gš,š(š„) = šŗ
š(š„) ,
limšā1
ā
Gš,š(š„, š¦) = šŗ
š(š„ + š¦) lim
šā1ā
gš,š= šŗš.
(18)
Here šµš(š„), šø
š(š„), and šŗ
š(š„) denote the classical Bernoulli,
Euler, and Genocchi polynomials, respectively, which aredefined by
š”
šš” ā 1šš”š„=
ā
ā
š=0
šµš(š„)
š”š
š!,
2
šš” + 1šš”š„=
ā
ā
š=0
šøš(š„)
š”š
š!,
2š”
šš” + 1šš”š„=
ā
ā
š=0
šŗš (š„)
š”š
š!.
(19)
The aim of the present paper is to obtain some resultsfor the above newly defined š-polynomials. It should bementioned that š-Bernoulli and š-Euler polynomials in ourdefinitions are polynomials of š„ and š¦ and when š¦ = 0,they are polynomials of š„. First advantage of this approach isthat for š ā 1ā, B
š,š(š„, š¦) (E
š,š(š„, š¦), G
š,š(š„, š¦)) becomes
the classical BernoulliBš(š„ + š¦) (EulerE
š(š„ + š¦), Genocchi
Gš,š(š„, š¦)) polynomial and wemay obtain the š-analogues of
well-known results, for example, Srivastava and PinteĢr [11],Cheon [23], and so forth. Second advantage is that, similarto the classical case, odd numbered terms of the Bernoullinumbers b
š,šand the Genocchi numbers g
š,šare zero, and
even numbered terms of the Euler numbers eš,š
are zero.
2. Preliminary Results
In this section we will provide some basic formulae for theš-Bernoulli, š-Euler, and š-Genocchi numbers and polyno-mials in order to obtain the main results of this paper in thenext section.
Lemma 5. The š-Bernoulli numbers bš,š
satisfy the followingš-binomial recurrence:
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšbš,šā bš,š= {
1, š = 1,
0, š > 1.(20)
Proof. By a simple multiplication of (8) we see that
BĢ (š”)Eš(š”) = š” + BĢ (š”) . (21)
Soā
ā
š=0
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšbš,š
š”š
[š]š!= š” +
ā
ā
š=0
bš,š
š”š
[š]š!. (22)
The statement follows by comparing š”š coefficients.
We use this formula to calculate the first few bš,š:
b0,š= 1,
b1,š= ā
1
2,
b2,š=1
4
š (š + 1)
š2 + š + 1=š[2]š
4[3]š
,
b3,š= 0.
(23)
The similar property can be proved for š-Euler numbers
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšeš,š+ eš,š
= {2, š = 0,
0, š > 0.(24)
and š-Genocchi numbers
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšgš,š+ gš,š
= {2, š = 1,
0, š > 1.(25)
Using the above recurrence formulae we calculate the firstfew eš,š
and gš,š
terms as well:
e0,š= 1, g
0,š= 0,
e1,š= ā
1
2, g
1,š= 1,
e2,š= 0, g
2,š= ā
[2]š
2= ā
š + 1
2,
e3,š=[3]š[2]š ā [4]š
8=š (1 + š)
8, g
3,š= 0.
(26)
4 Abstract and Applied Analysis
Remark 6. The first advantage of the new š-numbersbš,š, eš,š, and g
š,šis that similar to classical case odd num-
bered terms of the Bernoulli numbers bš,š
and the Genocchinumbers g
š,šare zero, and even numbered terms of the Euler
numbers eš,š
are zero.
Next lemma gives the relationship between š-Genocchinumbers and š-Tangent numbers.
Lemma 7. For any š ā N, we have
T2š+1,š
= g2š+2,š
(ā1)šā122š+1
[2š + 2]š
. (27)
Proof. First we recall the definition of š-trigonometric func-tions:
cosšš” =
šš(šš”) + š
š(āšš”)
2, sin
šš” =
šš(šš”) ā š
š(āšš”)
2š,
š tanšš” =
šš(šš”) ā š
š(āšš”)
šš (šš”) + šš (āšš”)
, cotšš” = š
šš(šš”) + š
š(āšš”)
šš (šš”) ā šš (āšš”)
.
(28)
Now by choosing š§ = 2šš” in BĢ(š§), we get
BĢ (2šš”) =2šš”
Eš(2šš”) ā 1
=š”šš(āšš”)
sinšš”
=
ā
ā
š=0
bš,š
(2šš”)š
[š]š!. (29)
It follows that
BĢ (2šš”) =š”šš(āšš”)
sinšš”
=š”
sinšš”(cosšš” ā š sin
šš”) = š” cot
šš” ā šš”
= b0,š+ 2šš”b
1,š+
ā
ā
š=2
bš,š
(2šš”)š
[š]š!
= 1 ā šš” +
ā
ā
š=2
bš,š
(2šš”)š
[š]š!.
(30)
Since the function š” cotšš” is even in the above sum odd
coefficients b2š+1,š
, š = 1, 2, . . ., are zero, and we get
š” cotšš” = 1 +
ā
ā
š=2
bš,š
(2šš”)š
[š]š!= 1 +
ā
ā
š=1
bš,š
(2šš”)2š
[2š]š!. (31)
By choosing š§ = 2šš” in GĢ(š§), we get
GĢ (2šš”) =4šš”
Eš(2šš”) + 1
=2šš”šš (āšš”)
cosšš”
=
ā
ā
š=0
gš,š
(2šš”)š
[š]š!,
GĢ (2šš”) =4šš”
Eš (2šš”) + 1
=2šš”šš (āšš”)
cosšš”
=2šš”
cosšš”(cosšš” ā š sin
šš”)
= 2šš” + 2š” tanšš” = g0,š+ 2šš”g
1,š+
ā
ā
š=2
gš,š
(2šš”)š
[š]š!
= 2šš” +
ā
ā
š=2
gš,š
(2šš”)š
[š]š!.
(32)
It follows that
2š” tanšš” =
ā
ā
š=1
g2š,š
(2šš”)2š
[2š]š!,
tanšš” =
ā
ā
š=1
g2š,š
(ā1)š(2š”)2šā1
[2š]š!,
tanhšš” = āš tan
š(šš”) = āš
ā
ā
š=1
g2š,š
(ā1)š(2šš”)2šā1
[2š]š!
= ā
ā
ā
š=1
g2š,š
(2š”)2šā1
[2š]š!= ā
ā
ā
š=1
g2š+2,š
(2š”)2š+1
[2š + 2]š!,
(33)
Thus
tanhšš” = āštan
š (šš”) =šš(š”) ā š
š(āš”)
šš(š”) + š
š(āš”)
=Eš(2š”) ā 1
Eš(2š”) + 1
=
ā
ā
š=1
T2š+1,š
(ā1)šš”2š+1
[2š + 1]š!,
T2š+1,š
= g2š+2,š
(ā1)šā122š+1
[2š + 2]š
.
(34)
The following result is a š-analogue of the additiontheorem, for the classical Bernoulli, Euler, and Genocchipolynomials.
Lemma 8 (addition theorems). For all š„, š¦ ā C we have
Bš,š(š„, š¦) =
š
ā
š=0
[š
š]
š
bš,š(š„ āšš¦)šāš
,
Bš,š(š„, š¦) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšBš,š (š„) š¦
šāš,
Abstract and Applied Analysis 5
Eš,š(š„, š¦) =
š
ā
š=0
[š
š]
š
eš,š(š„ āšš¦)šāš
,
Eš,š(š„, š¦) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšEš,š(š„) š¦šāš,
Gš,š(š„, š¦) =
š
ā
š=0
[š
š]
š
gš,š(š„ āšš¦)šāš
,
Gš,š(š„, š¦) =
š
ā
š=0
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšGš,š(š„) š¦šāš.
(35)
Proof. We prove only the first formula. It is a consequence ofthe following identity:
ā
ā
š=0
Bš,š(š„, š¦)
š”š
[š]š!=
š”
Eš(š”) ā 1
Eš(š”š„)E
š(š”š¦)
=
ā
ā
š=0
bš,š
š”š
[š]š!
ā
ā
š=0
(š„ āšš¦)š š”š
[š]š!
=
ā
ā
š=0
š
ā
š=0
[š
š]
š
bš,š(š„ āšš¦)šāš š”š
[š]š!.
(36)
In particular, setting š¦ = 0 in (35), we get the followingformulae for š-Bernoulli, š-Euler and š-Genocchi polynomi-als, respectively:
Bš,š(š„) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšbš,šš„šāš,
Eš,š(š„) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšeš,šš„šāš,
(37)
Gš,š (š„) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšgš,šš„šāš. (38)
Setting š¦ = 1 in (35), we get
Bš,š(š„, 1) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšBš,š(š„) ,
Eš,š(š„, 1) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšEš,š(š„) ,
Gš,š(š„, 1) =
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšGš,š(š„) .
(39)
Clearly (39) is š-analogues of
šµš (š„ + 1) =
š
ā
š=0
(š
š)šµš (š„) ,
šøš(š„ + 1) =
š
ā
š=0
(š
š)šøš(š„) ,
šŗš(š„ + 1) =
š
ā
š=0
(š
š)šŗš(š„) ,
(40)
respectively.
Lemma 9. The odd coefficients of the š-Bernoulli numbers,except the first one, are zero. That means b
š,š= 0 where
š = 2š + 1 (š ā N).
Proof. It follows from the fact that the function
š (š”) =
ā
ā
š=0
bš,š
š”š
[š]š!ā b1,šš”
=š”
Eš (š”) ā 1
+š”
2=š”
2(Eš (š”) + 1
Eš (š”) ā 1
) ,
(41)
By using š-derivative we obtain the next lemma.
Lemma 10. One has
š·š,š„Bš,š (š„) = [š]š
Bšā1,š (š„) +Bšā1,š (šš„)
2,
š·š,š„Eš,š(š„) = [š]š
Ešā1,š
(š„) + Ešā1,š
(šš„)
2,
š·š,š„Gš,š (š„) = [š]š
Gšā1,š
(š„) +Gšā1,š
(šš„)
2.
(42)
Lemma 11 (difference equations). One has
Bš,š(š„, 1) āB
š,š(š„) =
(ā1; š)šā1
2šā1[š]šš„šā1, š ā„ 1, (43)
Eš,š (š„, 1) + Eš,š (š„) = 2
(ā1; š)š
2šš„š, š ā„ 0, (44)
Gš,š(š„, 1) +G
š,š(š„) = 2
(ā1; š)šā1
2šā1[š]šš„šā1, š ā„ 1. (45)
Proof. Weprove the identity for the š-Bernoulli polynomials.From the identity
š”Eš(š”)
Eš(š”) ā 1
Eš (š”š„) = š”Eš (š”š„) +
š”
Eš(š”) ā 1
Eš (š”š„) , (46)
it follows thatā
ā
š=0
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšBš,š (š„)
š”š
[š]š!
=
ā
ā
š=0
(ā1, š)š
2šš„š š”š+1
[š]š!+
ā
ā
š=0
Bš,š(š„)
š”š
[š]š!.
(47)
6 Abstract and Applied Analysis
From (43) and (37), (44) and (38), we obtain the followingformulae.
Lemma 12. One has
š„š=
2š
(ā1; š)š[š]š
š
ā
š=0
[š + 1
š]
š
(ā1; š)š+1āš
2š+1āšBš,š(š„) ,
š„š=
2šā1
(ā1; š)š
(
š
ā
š=0
[š
š]
š
(ā1; š)šāš
2šāšEš,š (š„) + Eš,š (š„)) ,
š„š=
2šā1
(ā1; š)š[š + 1]š
Ć (
š+1
ā
š=0
[š + 1
š]
š
(ā1; š)š+1āš
2š+1āšGš,š(š„) +G
š+1,š(š„)) .
(48)
The above formulae are š-analogues of the followingfamiliar expansions:
š„š=
1
š + 1
š
ā
š=0
(š + 1
š)šµš(š„) ,
š„š=1
2[
š
ā
š=0
(š
š)šøš (š„) + šøš (š„)] ,
š„š=
1
2 (š + 1)[
š+1
ā
š=0
(š + 1
š)šøš(š„) + šø
š+1(š„)] ,
(49)
respectively.
Lemma 13. The following identities hold true:
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšBš,š(š„, š¦) āB
š,š(š„, š¦)
= [š]š(š„ āš š¦)šā1
,
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšEš,š(š„, š¦) + E
š,š(š„, š¦) = 2(š„ ā
šš¦)š
,
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšGš,š(š„, š¦) +G
š,š(š„, š¦)
= 2[š]š(š„ āš š¦)šā1
.
(50)
Proof. We prove the identity for the š-Bernoulli polynomi-als. From the identity
š”Eš(š”)
Eš(š”) ā 1
Eš (š”š„)Eš (š”š¦)
= š”Eš(š”š„)E
š(š”š¦) +
š”
Eš (š”) ā 1
Eš(š”š„)E
š(š”š¦) ,
(51)
it follows thatā
ā
š=0
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšBš,š(š„, š¦)
š”š
[š]š!
=
ā
ā
š=0
š
ā
š=0
[š
š]
š
(ā1, š)š(ā1, š)
šāš
2šš„šš¦šāš š”š+1
[š]š!
+
ā
ā
š=0
Bš,š(š„, š¦)
š”š
[š]š!.
(52)
3. Some New Formulae
The classical Cayley transformation š§ āCay(š§, š) := (1 +šš§)/(1āšš§)motivated us to derive the formula forE
š(šš”). In
addition, by substituting Cay(š§, (š ā 1)/2) in the generatingformula we have
BĢš(šš”) BĢ
š (š”) = (BĢq (šš”) ā šBĢš (š”) (1 + (1 ā š)š”
2))
Ć1
1 ā šĆ
2
Eš (š”) + 1
.
(53)
The right hand side can be presented by š-Euler numbers.Now equating coefficients of š”š we get the following identity.In the case that š = 0, we find the first improved š-Eulernumber which is exactly 1.
Proposition 14. For all š ā„ 1,š
ā
š=0
[š
š]
š
Bš,šBšāš,š
šš
= āš
š
ā
š=0
[š
š]
š
(ā1, š)šāš
2šāšBš,šEšāš,š[š ā 1]š
āš
2
šā1
ā
š=0
[š ā 1
š]
š
(ā1, š)šāšā1
2šāšā1Bš,šEšāšā1,š[š]š.
(54)
Let us take a š-derivative from the generating function,after simplifying the equation, by knowing the quotient rulefor quantum derivative, and also using
Eš(šš”) =
1 ā (1 ā š) (š”/2)
1 + (1 ā š) (š”/2)Eš(š”) ,
š·š(Eq (š”)) =
Eš(šš”) +E
š (š”)
2,
(55)
one has
BĢš(šš”) BĢ
š(š”) =
2 + (1 ā š) š”
2Eš (š”) (š ā 1)
(šBĢš(š”) ā BĢ
š(šš”)) . (56)
It is clear that Eā1š(š”) = E
š(āš”). Now, by equating
coefficients of š”š we obtain the following identity.
Abstract and Applied Analysis 7
Proposition 15. For all š ā„ 1,
2š
ā
š=0
[2š
š]
š
Bš,šB2šāš,š
šš
= āš
2š
ā
š=0
[2š
š]
š
(ā1, š)2šāš
22šāšBš,š[š ā 1]š(ā1)
š
+š (1 ā š)
2
2šā1
ā
š=0
[2š ā 1
š]
š
(ā1, š)2šā1āš
22šā1āšBš,š
Ć [š ā 1]š(ā1)š,
2š+1
ā
š=0
[2š + 1
š]
š
Bš,šB2šāš+1,š
šš
= š
2š+1
ā
š=0
[2š + 1
š]
š
(ā1, š)2š+1āš
22š+1āšBš,š[š ā 1]š(ā1)
š
āš (1 ā š)
2
2š
ā
š=0
[2š
š]
š
(ā1, š)2šāš
22šāšBš,š[š ā 1]š(ā1)
š.
(57)
We may also derive a differential equation for BĢš(š”). If
we differentiate both sides of the generating function withrespect to š”, after a little calculation we find that
š
šš”BĢš(š”)
= BĢš (š”) (
1
š”ā(1 ā š)E
š(š”)
Eš(š”) ā 1
(
ā
ā
š=0
4šš
4 ā (1 ā š)2š2š
)) .
(58)
If we differentiate BĢš(š”) with respect to š, we obtain,
instead,
š
ššBĢš(š”) = āBĢ
2
š(š”)Eš(š”)
ā
ā
š=0
4š” (šššā1
ā (š + 1) šš)
4 ā (1 ā š)2š2š
. (59)
Again, using the generating function and combining thiswith the derivative we get the partial differential equation.
Proposition 16. Consider the following:
š
šš”BĢš(š”) ā
š
ššBĢš(š”)
=BĢš(š”)
š”+
BĢ2š(š”)Eš(š”)
š”
Ć
ā
ā
š=0
4š” (šššā1
ā (š + 1) šš) ā šš(1 ā š)
4 ā (1 ā š)2š2š
.
(60)
4. Explicit Relationship between theš-Bernoulli and š-Euler Polynomials
In this section, we give some explicit relationship betweenthe š-Bernoulli and š-Euler polynomials.We also obtain newformulae and some special cases for them.These formulae areextensions of the formulae of Srivastava and PinteĢr, Cheon,and others.
We present natural š-extensions of themain results in thepapers [9, 11]; see Theorems 17 and 19.
Theorem 17. For š ā N0, the following relationships hold true:
Bš,š(š„, š¦)
=1
2
š
ā
š=0
[š
š]
š
ššāš [
[
Bš,š (š„) +
š
ā
š=0
[š
š]
š
(ā1, š)šāš
Bš,š (š„)
2šāšššāš]
]
Ć Ešāš,š
(šš¦)
=1
2
š
ā
š=0
[š
š]
š
ššāš
[Bš,š(š„) +B
š,š(š„,
1
š)]Ešāš,š
(šš¦) .
(61)
Proof. Using the following identity
š”
Eš(š”) ā 1
Eš(š”š„)E
š(š”š¦)
=š”
Eš(š”) ā 1
Eš (š”š„)
ā Eš (š”/š) + 1
2ā
2
Eš(š”/š) + 1
Eš(š”
ššš¦)
(62)
we have
ā
ā
š=0
Bš,š(š„, š¦)
š”š
[š]š!
=1
2
ā
ā
š=0
Eš,š(šš¦)
š”š
šš[š]š!
ā
ā
š=0
(ā1, š)š
šš2š
š”š
[š]š!
ā
ā
š=0
Bš,š(š„)
š”š
[š]š!
+1
2
ā
ā
š=0
Eš,š(šš¦)
š”š
šš[š]š!
ā
ā
š=0
Bš,š (š„)
š”š
[š]š!
=: š¼1+ š¼2.
(63)
It is clear that
š¼2=1
2
ā
ā
š=0
Eš,š(šš¦)
š”š
šš[š]š!
ā
ā
š=0
Bš,š(š„)
š”š
[š]š!
=1
2
ā
ā
š=0
š
ā
š=0
[š
š]
š
ššāš
Bš,š (š„)Ešāš,š (šš¦)
š”š
[š]š!.
(64)
8 Abstract and Applied Analysis
On the other hand
š¼1=1
2
ā
ā
š=0
Eš,š(šš¦)
š”š
šš[š]š!
Ć
ā
ā
š=0
š
ā
š=0
[š
š]
š
Bš,š (š„)
(ā1, š)šāš
ššāš2šāš
š”š
[š]š!
=1
2
ā
ā
š=0
š
ā
š=0
[š
š]
š
Ešāš,š
(šš¦)
Ć
š
ā
š=0
[š
š]
š
Bš,š(š„) (ā1, š)
šāš
ššāšššāš2šāš
š”š
[š]š!.
(65)
Thereforeā
ā
š=0
Bš,š(š„, š¦)
š”š
[š]š!
=1
2
ā
ā
š=0
š
ā
š=0
[š
š]
š
ššāš
Ć [
[
Bš,š (š„) +
š
ā
š=0
[š
š]
š
(ā1, š)šāš
Bš,š(š„)
2šāšššāš]
]
Ć Ešāš,š
(šš¦)š”š
[š]š!.
(66)
It remains to equate the coefficients of š”š.
Next we discuss some special cases of Theorem 17.
Corollary 18. For š ā N0the following relationship holds true:
Bš,š(š„, š¦)
=
š
ā
š=0
[š
š]
š
(Bš,š (š„) +
(ā1; š)šā1
2š[š]šš„šā1)Ešāš,š
(š¦) .
(67)
The formula (67) is a š-extension of the Cheonās mainresult [23].
Theorem 19. For š ā N0, the following relationships
Eš,š(š„, š¦)
=1
[š + 1]š
Ć
š+1
ā
š=0
1
šš+1āš[š + 1
š]
š
Ć (
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšEš,š(š¦) ā E
š,š(š¦))
ĆBš+1āš,š (šš„)
(68)
hold true between the š-Bernoulli polynomials and š-Eulerpolynomials.
Proof. The proof is based on the following identity:2
Eš(š”) + 1
Eš (š”š„)Eš (š”š¦)
=2
Eš (š”) + 1
Eš(š”š¦)
ā Eš(š”/š) ā 1
š”ā
š”
Eš(š”/š) ā 1
Eš(š”
ššš„) .
(69)
Indeedā
ā
š=0
Eš,š(š„, š¦)
š”š
[š]š!
=
ā
ā
š=0
Eš,š(š¦)
š”š
[š]š!
ā
ā
š=0
(ā1, š)š
šš2š
š”šā1
[š]š!
Ć
ā
ā
š=0
Bš,š(šš„)
š”š
šš[š]š!
ā
ā
ā
š=0
Eš,š(š¦)
š”šā1
[š]š!
ā
ā
š=0
Bš,š(šš„)
š”š
šš[š]š!
=: š¼1ā š¼2.
(70)
It follows that
š¼2=1
š”
ā
ā
š=0
Eš,š(š¦)
š”š
[š]š!
ā
ā
š=0
Bš,š(šš„)
š”š
šš[š]š!
=1
š”
ā
ā
š=0
š
ā
š=0
[š
š]
š
1
ššāšEš,š(š¦)B
šāš,š (šš„)š”š
[š]š!
=
ā
ā
š=0
1
[š + 1]š
Ć
š+1
ā
š=0
[š + 1
š]
š
1
šš+1āšEš,š(š¦)B
š+1āš,š (šš„)š”š
[š]š!,
š¼1=1
š”
ā
ā
š=0
Bš,š(šš„)
š”š
šš[š]š!
Ć
ā
ā
š=0
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšEš,š(š¦)
š”š
[š]š!
=1
š”
ā
ā
š=0
š
ā
š=0
[š
š]
š
1
ššāšBšāš,š
(šš„)
Ć
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšEš,š(š¦)
š”šā1
[š]š!.
(71)
Next we give an interesting relationship between the š-Genocchi polynomials and the š-Bernoulli polynomials.
Abstract and Applied Analysis 9
Theorem 20. For š ā N0, the following relationship
Gš,š(š„, š¦)
=1
[š + 1]š
Ć
š+1
ā
š=0
1
ššāš[š + 1
š]
š
Ć (
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšGš,š(š„) āG
š,š(š„))
ĆBš+1āš,š
(šš¦) ,
Bš,š(š„, š¦)
=1
2[š + 1]š
Ć
š+1
ā
š=0
1
ššāš[š + 1
š]
š
Ć (
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšBš,š(š„) +B
š,š(š„))
ĆGš+1āš,š
(šš¦)
(72)
holds true between the š-Genocchi and the š-Bernoulli polyno-mials.
Proof. Using the following identity
2š”
Eš(š”) + 1
Eš(š”š„)E
š(š”š¦)
=2š”
Eš (š”) + 1
Eš(š”š„) ā (E
š(š”
š) ā 1)
š
š”
ā š”/š
Eš(š”/š) ā 1
ā Eš(š”
ššš¦)
(73)
we have
ā
ā
š=0
Gš,š(š„, š¦)
š”š
[š]š!
=š
š”
ā
ā
š=0
Gš,š(š„, š¦)
š”š
[š]š!
Ć
ā
ā
š=0
(ā1, š)š
šš2š
š”š
[š]š!
ā
ā
š=0
Bš,š(šš¦)
š”š
šš[š]š!
āš
š”
ā
ā
š=0
Gš,š(š„, š¦)
š”š
[š]š!
ā
ā
š=0
Bš,š(šš¦)
š”š
šš[š]š!
=š
š”
ā
ā
š=0
(
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšGš,š (š„) āGš,š (š„))
š”š
[š]š!
Ć
ā
ā
š=0
Bš,š(šš¦)
š”š
šš[š]š!
=š
š”
ā
ā
š=0
š
ā
š=0
1
ššāš[š
š]
š
Ć (
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšGš,š(š„) āG
š,š(š„))
ĆBšāš,š
(šš¦)š”š
[š]š!
=
ā
ā
š=0
1
[š + 1]š
š+1
ā
š=0
1
ššāš[š + 1
š]
š
Ć (
š
ā
š=0
[š
š]
š
(ā1, š)šāš
ššāš2šāšGš,š(š„) āG
š,š(š„))
ĆBš+1āš,š
(šš¦)š”š
[š]š!.
(74)
The second identity can be proved in a like manner.
Conflict of Interests
The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper.
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