STUDY ON CLASS OF IMPROVED Q-BERNOULLI

MATRIX AND ITS PROPERTIES

A THESIS SUBMITTED TO THE GRADUATE

SCHOOL OF APPLIED SCIENCES

OF

NEAR EAST UNIVERSITY

By

IBRAHIM YUSUF KAKANGI

In Partial Fulfilment of the Requirements for

the Degree of Master of Science

in

Mathematics

NICOSIA, 2017

IBR

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STUDY ON CLASS OF IMPROVED Q-BERNOULLI

MATRIX AND ITS PROPERTIES

A THESIS SUBMITTED TO THE GRADUATE

SCHOOL OF APPLIED SCIENCES

OF

NEAR EAST UNIVERSITY

By

IBRAHIM YUSUF KAKANGI

In Partial Fulfilment of the Requirements for

the Degree of Master of Science

in

Mathematics

NICOSIA, 2017

Ibrahim Yusuf KAKANGI: STUDY ON CLASS OF IMPROVED Q-BERNOULLI

MATRIX AND ITS PROPERTIES

Approval of Director of Graduate School of

Applied Sciences

Prof. Dr. Nadire Çavuş

We certify that, this thesis is satisfactory for the award of the degree of Master of

Sciences in Mathematics.

Examining Committee in Charge:

Prof. Dr. Allaberen Ashyralyev Committee Chairman, Department of

Mathematics, Near East University.

Assoc.Prof. Dr. Suzan Cival Buranay External Examiner, Department of

Mathematics, Eastern Mediterranean

University.

Assis. Prof. Dr. Mohammad Momenzadeh Supervisor, Department of

Mathematics, Near East University.

I hereby declare that all information in this document has been obtained and presented in

accordance with academic rules and ethical conduct. I also declare that, as required by these

rules and conduct, I have fully cited and referenced all material and results that are not original

to this work.

Name, Last name: Yusuf Ibrahim Kakangi

Signature:

Date

ACKNOWLEDGMENTS

My deepest appreciation goes to Almighty God for being my strength, help and my reference

point towards the completion of my master degree course.

I wish to express my sincere gratitude to my supervisor Ass.Prof.Dr. Mohammad

MOMENZADEH invaluable assistance, guidance and thorough supervision. His keen eyes for

details and uncompromising insistence on high standard has ensured the success of this thesis.

I am most indebted to my sponsors: Kaduna State Government, Nigeria. And I am most

grateful to my tireless parent, brother, sisters, Nephew, cousins, relatives, friends and course

mate, for their support, useful advice, and encouragement towards the completion of my

master’s programme. May almighty Allah bless you and grant all your heart desires.

To Dr. Ramalan Yero ….

i

ABSTRACT

Since 19th century, a lot of q-Bernoulli numbers and polynomials has been introduced. Carlitz

was the first who made a generation of q-Bernoulli numbers, afterwards, a lot of researcher’s

works on a new form of q-Bernoulli numbers and matrices. In this thesis, we introduce

ordinary Bernoulli and q-Bernoulli matrices and their related Pascal matrices and their

relations. At the end by using generating function and improved q-exponential function we

work on a new class of q-Bernoulli matrix and related properties are given. Our definition is

more significant since it demonstrates a better definition of q-Bernoulli matrix and the

properties are convinced the ordinary case as well.

KEYWORDS: Bernoulli Number; Bernoulli Matrices; q-Bernoulli Number; q-Bernoulli

Matrices; Improved q-Bernoulli number; and Improved q-Bernoulli Matrices

ii

ÖZET

19. yüzyıldan beri, bir sürü q-Bernoulli sayısı ve polinomları tanıtıldı. Carlitz, daha sonra, q-

Bernoulli sayılarının ve matrislerinin yeni bir formuyla ilgili birçok araştırmacı tarafından q-

Bernoulli sayıları üreten ilk kişiydi. Bu tezde sıradan Bernoulli ve q-Bernoulli matrislerini ve

ilgili Pascal matrislerini ve bunların ilişkilerini tanıtmaktayız. Sonunda üretme fonksiyonu ve

geliştirilmiş q-üstel fonksiyonu kullanılarak q-Bernoulli matrisinin yeni bir sınıfında çalıştık

ve ilgili özellikler verildi. Tanımımız, q-Bernoulli matrisinin daha iyi tanımlanmasını

gösterdiği için daha belirgindir ve özellikler olağan durumu ikna eder.

ANAHTAR KELİMELER: Bernoulli sayısı; Bernoulli matrisleri; q-Bernoulli sayısı; q-

Bernoulli matrisleri; Geliştirilmiş q-Bernoulli sayısı; ve

Geliştirilmiş q-Bernoulli Matrisleri

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS………………………………………………………………... i

ABSTRACT……………………………………………………………………………….. ii

ÖZET………………………………………………………………………………………. iii

TABLE OF CONTENTS………………………………………………………………… iii

CHAPTER 1: INTRODUCTION…………………………………………………………… 1

1.1 Quantum Calculus.........................................................................................................1

1.1.1 Definition q And h-Differentiation............................................................................2

1.1.2 Definition q And h-Derivative..................................................................................2

1.1.3 Lemma Linearity Of q And h-Derivative..................................................................3

1.2 q-Taylor’s Formula For Polynomial.............................................................................4

1.2.1 q-Analogue Of Some q-Combinatory.........................................................................4

1.2.2 Some Properties of q-Calculus Functions..................................................................5

1.3 q-Exponential Function.................................................................................................6

1.3.1 Gauss binomial formula..............................................................................................6

1.3.2 Heines Binomial Formula...........................................................................................6

1.4.3 q-Euler Identities.........................................................................................................7

1.3.4 q-Exponential Functions..............................................................................................8

1.3.5 Relationship Between eqx and Eqx...........................................................................8

1.3.6 q-Derivative Of The q-Exponential Functions............................................................9

1.3.7 Convergence Of q-Exponentials Functions.................................................................9

1.4 q-Trigonometric Functions.........................................................................................10

iv

1.4.1 Properties Of q-Trigonometric Functions.................................................................10

1.4.2 q-Derivetive Of q-Trigonometric Functions.............................................................10

1.5 Improved q-Fxponential Function..............................................................................11

1.5.1 Definition ℇ qz.........................................................................................................11

1.5.2 Basic Definitions on Improved q-Exponential Function...........................................11

1.5.3 Unification Of q-Exponential Functions.................................................................12

1.5.4 Improved q-Trigonometric Functions......................................................................13

1.6 Bernoulli Number.......................................................................................................13

1.6.1 Recurrence Formula for Ordinary Bernoulli Number...............................................14

1.6.2 Kronecker Delta.......................................................................................................15

1.6.3 Lemma Explicit Definition of Bernoulli Number....................................................16

1.6.4 Proposition Bernoulli Numbers as Rational numbers............................................16

1.6.5 Bernoulli Polynomials.............................................................................................16

1.6.6 Some Properties Of Bernoulli Polynomials............................................................17

CHAPTER 2: BERNOULLI MATRIX AND SOME PROPERTIES…………………... 20

2.1 Bernoulli Matrix..........................................................................................................20

2.1.1 Definition Ordinary Matrix and some Properties...................................................20

2.1.2 Definition Bernoulli Matrix and Bernoulli Polynomials..........................................21

2.1.3 Theorem Bernoulli Polynomial Matrix of x and y..................................................21

2.1.4 Definition Inverse Of Bernoulli Matrix..................................................................24

2.2 Bernoulli Matrix and Generalized Pascal Matrix.......................................................25

2.2.1 Definition Pascal Matrix........................................................................................25

2.2.2 Theorem Relationship Between Barnoulli Polynomial Matrix And Pascal Matrix 25

2.2.3 Theorem Inverse Of Bernoulli Polynomial Matrix and Pascal Matrix....................28

v

CHAPTER 3: Q-BERNOULLI MATRIX AND ITS PROPERTIES…………………… 32

3.1 Q-Bernoulli Matrix.....................................................................................................32

3.1.1 Definition q-Bernoulli numbers...............................................................................32

3.1.2 Definition q-Bernoulli Polynomials.........................................................................33

3.1.3 Definition q-Bernoulli Matrix..................................................................................33

3.1.4 Definition q-Bernoulli Polynomials Matrix.............................................................34

3.1.5 Theorem Inverse of q-Bernoulli Matrix...................................................................34

3.2 q-Bernoulli Matrix and q-Pascal Matrices.......................................................................36

3.2.1 Definition Pascal Matrix and Inverse of Pascal Matrix...........................................36

3.2.2 Theorem Relationship Between q-Bernoulli Polynomial Matrix and Pascal Matrix

............................................................................................................................................37

3.2.3 Definition Inverse of q-Bernoulli Polynomials Matrix............................................41

3.2.4 Corollary......................................................................................................................42

CHAPTER 4: IMPROVED Q-BERNOULLI MATRIX AND ITS PROPERTIES……. 44

4.1 History of q-Bernoulli Numbers......................................................................................44

4.1.1 Definition Carlitz q-Bernoulli Number....................................................................44

4.2 Improved q-Bernoulli Numbers......................................................................................45

4.2.1 Lemma Recurrence Formula For Improved q-Bernoulli Number...........................45

4.2.3 Lemma Advantage Of Improved q-Exponential Function......................................46

4.2.4 Improved q-Bernoulli Polynomials...........................................................................48

4.2.5 Theorem Additive Theory........................................................................................48

4.3 Q-Improved Bernoulli Matrix And Its Properties..........................................................51

vi

4.3.1 Definition Improved q-Bernoulli Matrix and Improved q-Bernoulli Polynomials

Matrix.................................................................................................................................51

4.3.2 Theorem Improved q-Bernoulli Polynomial Matrix in terms of x and y................51

4.3.3 Theorem Inverse of Improved q-Bernoulli matrix………………………………. 52

CHAPTER 5: SUMMARY AND CONCLUSION………………………………………. 54

REFERENCE……………………………………………………………………………… 54

vii

CHAPTER 1

INTRODUCTION

Majority of the work in this chapter was presented from The book ‘ A comprehensive

treatment of q-Calculus’ (Ernst.T, 2012) , ‘Quantum Calculus’ (Kac.V, and Cheung.P,2002)

and any other information that is not from there was cited by means of reference.

Among the most important sequence in mathematics is the sequence of Bernoulli numbers Bn

it has a quiet good relationship to the number theories, for example you can express the value

of 𝜁(2𝑛) using the Bernoulli number, where 𝑛 is a positive integer and 𝜁(𝑛) is a Riemann

zeta function [1] you can also find the uses of Bernoulli number in analysis, for instance, they

also find it in the Euler-Maclaurins formula, the formula that is very useful in physics and

mathematics, in asymptotic of a q-special functions, the Bernoulli numbers is very essential.

Bernoulli matrix, Pascal matrix are some example of matrix with binomial coefficients as their

element which are very important in matrix theory and combinatory. So many researchers has

been showing interest in this related area, for that reason we also want to relate this kind of

matrix with quantum calculus (q-calculus), but just before then here are some terminologies

that one needs to know about q-calculus.

1.1 Quantum Calculus

If

limx→ x0

f ( x )− f (x0)x−x0

(1.1)

exists, that gives us the well known definition of the derivative dydx of a function f(x) at a point

x=x0. However if we assume that x=q x0 or x=x0+h, where q is a fixed constant not equal to

1 and h is a fixed constant not equal to 0, and do not take the limit, we enter into different

concept of mathematics called the quantum calculus.

1

Quantum calculus involves two types of derivative which are; q-derivative and h-derivative

that leads to the study of q-calculus and h-calculus respectively. In the course of studying

quantum calculus in relation to the ordinary calculus, so many important results and notions in

number theory, combinatory and different area of mathematics have been discovered.

For instance, a q-derivative ofxn=[ n ]q xn−1 , where

[ n ]q=qn−1q−1

(1.2)

and [ n ]q represent the ordinary 'n' in the ordinary derivative of xn.

1.1.1 Definition q and h-Differentiation

A q-differential and h-differential of an arbitrary function say f (x) on the set of real numbers

are defined as (Kac.V, and Cheung.P,2002)

dq f (x)=f (qx )−f ( x )(1.3)

And

dh f ( x )=f (x+h )− f ( x )(1.4)

Respectively.

Particularly keeping in mind that, dq x=(q−1 ) x also that dh x=h.

Considering the q-differential and the h-differential we defined their corresponding quantum

derivative as

1.1.2 Definition q and h-Derivative

Supposed f ( x ) is an arbitrary function on the set of real numbers R. Its q-derivative and h-

derivative are defined as

Dq f ( x )=dq f ( x )

dq x=

f (qx )−f ( x )(q−1 ) x

(1.5)

where, q ≠ 1 ,∧x ≠0

Dh f ( x )=dh f (x )

dh x=

f (x+h )−f (x)h

(1.6)

2

with h ≠ 0

referred to as the q-derivative and h-derivative respectively, of the arbitrary function f (x)

If we noticed that

limq→1

D q f (x )=limh→ 0

D h f (x)=df (x)dx

provided that the function f ( x ) is differentiable. Looking at notation of Leibniz df ( x )

dx, which

has to do with the ratio of two ‘infinitesimals’ is somewhat difficult to understand, because

there is need to give further detail of the notion of the differential df ( x ). But on the other hand,

one can easily see on the notion of q-calculus and h- calculus also the q-derivative and the h-

derivative are plain ratios.

1.1.3 Lemma Linearity of q and h-Derivative

Just like in the concept of ordinary derivative, the linear operator behaves in the same way

while finding the q-derivative or h-derivative of a function. In essence, if Dq and Dh are q-

derivative and h-derivative, then for any constants a and b, the following property hold:

Dq (af ( x )+bg ( x ) )=a D q f (x )+b D q g ( x )(1.7)

Dh (af ( x )+bgf (x ) )=a Dh f ( x )+b Dh g ( x )(1.8)

Example: if f ( x )=xn, and n is an integer greater than zero, then the q-derivative and h-

derivative can be find as

Dq xn=qn xn−xn

qx−x=

(qn−1)xn

(q−1)x=

(qn−1)(q−1)

xn−1(1.9)

Dh xnh=

(x+h )n+xn

h=nxn−1+

n (n−1 ) xn−2 h2

+…+hn−1(1. 10)

but there is a frequent appearance of (q¿¿n−1)

(q−1)¿ in the q-derivative so therefore we used the

notation

[ n ]q=(q¿¿ n−1)

(q−1)=qn−1+…+1 ¿

3

And it is referred to as the q-analogue of n, for any integer n greater than zero, and hence (1.9)

becomes

Dq xn=[n ]q xn−1(1.11)

1.2 q-Taylor’s Formula For Polynomial

Before going to q-Taylor formula, lets recall the generalized Taylor formula in the ordinary

calculus.

Taylor theorem says

f ( x )=∑n=0

∞

f (n)(a)(x−a)n

n !,(1.12)

is the power series of any function f ( x ) which has derivative of all kind of order is analytic at

x=a, provided we can write it as a power series about a point x=a.

We can increase the definition of a function to a more interesting domain by Taylor expansion

of an analytic function. For instance, if we defined the exponentials as a square matrices and a

complex number by using the Taylor expansion of ex, with which then we express the q-

analogue of the following expression where the q-Taylor formula follows

1.2.1 q-Analogue of Some q-Combinatory

Definition 1.2.1 (Kac.V, and Cheung.P,2002) If n is a positive integer. we defined the q-

analogue of n ! as:

[n]q !={ 1 , for n=0[n ] [ n−1 ] !q , for n ≥1

(1.13)

Definition 1.2.2 The q- binomial coefficients of any integer n ≥ k ≥1, is defined as

(nk)q=

[n ] !q

[ k ] !q [n−k ] !q,(1.14)

and (1.13) satisfies (n0)q=1 also

(nk)q=0 for n<k (Naim and KUS, 2015)

4

(nk)q=( n

n−k )q,(1.15)

and

(nk)q(k

j)q=(nj)q

(n− jk− j)q

(1.16)

the q-analogue of binomial function (x−a)n is defined as:

(x−a)qn={ 1 , for n=0

( x−a ) ( x−aq ) …( x−aqn−1) , for n=1,2 , …(1.17)

Definition 1.2.3 (Kac.V, and Cheung.P,2002)

For n<0, q-analogue of (x−a)−n is defined as:

( x−a )q−n= 1

( x−q−n a )qn , for n=1,2 , …(1.18)

Definition 1.2.4 (Kac.V, and Cheung.P,2002)

Letα∈Z , the q-analogue of α is defined as:

[ α ]q=1−qα

1−q(1.19)

Definition 1.2.5 (Kac.V, and Cheung.P,2002)

Generalize q-polynomial function is defined as

pn ( x )=(x−a)q

n

[n ] !q(1.20)

Where pn(x ) is a polynomial.

1.2.2 Some Properties of q-Calculus Functions

Proposition 1.2.1 (Kac.V, and Cheung.P,2002)

The following properties hold for any integerm , n∈Z .

a. Dq ( x−a )qn=[ n ]q ( x−a )q

n−1

b. ( x−a )qm+n ≠ ( x−a )q

m ( x−a )qn .

5

c. ( x−a )qm+n=( x−a )q

m ( x−qm a )qn .

d. Dq( 1( x−a )q

n )=[−n ]q ( x−qna )q−n−1

.

e. (−1 )n qn ( n−1)

2 ( x−q−n+1 a )qn=(a−x )q

n .

f. Dq (a−x )qn=[−n ]q (a−qx )q

n−1 .

g. Dq( 1(a−x)q

n )= [ n ]q(a−x)q

n+1 .

By using the above definitions and proposition we eventually come up with the q-Taylor

binomial formula for polynomial (Kac.V, and Cheung.P,2002) as

f ( x )=∑j=0

N

(D¿¿q j f )(c)( x−c)q

j

[ j ] !q(1.21)¿

1.3 q-Exponential Function

Before we study the Euler identity and the q-exponential function, there is need to understand

the concept of Gauss’s binomial formula and Hein’s binomial formula which were both

derived from the q-Taylor binomial formula, in this case we assumed that f ( x )=(x−a)qn.

With x as a variable and using (1.17) of definition 1.2.2 we obtain the Gauss binomial formula

1.3.1 Gauss binomial formula

(x+a)qn=∑

j=0

n

[nk ]qq

(n2)

ak xn−k (1.22)

Where,

[nk ]q=

[ n ]q [n−1 ]q … [n+k−1 ]q[ k ] !q

=[ n ] !q

[ k ] !q [ n−k ] !q

Is the q-binomial coefficient

And the Heine’s binomial formula

1.3.2 Heine’s Binomial Formula

6

1(1−x )q

n =1+∑k=1

∞ [ n ]q [n+1 ]q … [ n+k−1 ]q[ k ] !q

xk(1.23)

(Kac.V, and Cheung.P,2002)

Now considering (1.22) by replacing x and a by 1 and x respectively i.e

(1+x)qn=∑

k=0

n

q(n2 )[nk ]

qx

k

and (1.22)

1(1−x )q

n =∑j=1

∞ [ n ]q [n+1 ]q… [ n+k−1 ]q[k ] !q

xk ,

What will happen if we take the limit of n as n → ∞ in both the expression? Depending on the

value of x, the result is infinitely small or infinitely large so therefore producing not

interesting result in the ordinary calculus i.e when q=1. But in q-calculus it is entirely

different because, an example is, assuming |q|<1 , the infinite product

(1+x)q∞=(1+x ) (1+qx ) (1+q2 x ) …

will eventually converge to some finite limit. Therefore if we let |q|<1 , we have

limn→ ∞

[ n ]q= limn → ∞

1−qn

1−q= 1

1−q(1.24)

and

limn → ∞ (n

k)=limn →∞

(1−qn ) (1−qn−1 ) … (1−qn−k +1 )(1−q)¿¿

¿

¿ 1(1−q)(1−q2)…(1−qk)

(1.25)

So therefore there is difference in the behaviour between the q-analogues of integer and

binomial coefficients for a n larger integer to their ordinary counterparts.

Taking the limits as n → ∞ and substituting (1.24) and (1.25) in the Heine’s and

Gauss’s binomial formula we develop two identities of formal power series in x (with the

assumption that |q|<1∨¿). (Kac.V, and Cheung.P,2002)

1.4.3 q-Euler Identities

(1+x )q∞=∑

k=0

∞

qk ( k−1)

2 xk

(1−q ) (1−q2 ) …( 1−qk )(1.26)

7

1(1−x )q

∞ =∑k=0

∞ xk

(1−q)(1−q2)… (1−qk )(1.27)

and call (1.26) and (1.27) Euler’s first and second identities or E1 and E2 respectively (Kac.V,

and Cheung.P,2002) because he was the one that reveals them at the time of his live before

Gauss’s and Heine. Also the identities relate infinite product and infinite sums but they don’t

have classical analogue because each and every term in the sum don’t have meaning when

q=1.

1.3.4 q-Exponential Functions

Studying those identities helps us to define the q-analogue of the exponential function, but

before then, lets recall the Taylors’s exponential function expansion. i.e

ex=∑k=0

∞ xk

k !(1.28)

From (1.27) dividing both the numerator and the denominator of the R.H.S by 1−q we got

∑k=0

∞ xk

1 (1−q2)1−q

… (1−qk)1−q

=∑k=0

∞ ( x1−q )

k

[ k ] !q(1.29)

Definition: (Kac.V, and Cheung.P,2002) The classical exponential function ex has a q-

analogue as

eqx=∑

k=0

∞ xk

[k ] !q(1.30)

By using (1.29) and (1.30) we get

eqx /(1−q)= 1

(1−x)q∞ ,

or its equivalent

eqx= 1

(1−(1−q) x)q∞ (1.31)

That is the case of E2, we can also use E1 to defined another q-exponential function.

Definition (Kac.V, and Cheung.P,2002)

8

Eqx=∑

k=0

∞

qk (k−1 )

2 xk

[ k ] !q=(1+ (1−q ) x)q

∞ (1.32)

We can relate (1.31) and (1.32) as

1.3.5 Relationship Between eqx and Eq

x

eqx Eq

− x=1(1.33)

From the above property we can say

e1/ qx =∑

k=0

∞ (1−1/q )k xk

(1−1/q ) (1−1/q2 ) … (1−1/qk )

¿∑k=0

∞

qk (k−1)/2 (1−q ) k xk

(1−q ) (1−q2 ) … (1−qk )=Eq

x (1.34)

1.3.6 q-Derivative Of The q-Exponential Functions

And the q-derivative of the two q-exponential function is given as

Dq eqx=∑

k=0

∞ Dq xk

[ k ] !q=∑

k=0

∞ [ k ]q xk−1

[k ] !q=∑

k =0

∞ xk−1

[ k−1 ] !q=¿eq

x (1.35)¿

Dq Eqx=∑

k=0

∞

qk (k−1 )

2 Dq xk

[ k ] !q=∑

k=0

∞

qk ( k−1 )

2 [ k ]q xk−1

[k ] !q=∑

k=0

∞

qk ( k−1)

2 xk−1

[ k−1 ] !q=Eq

qx(1.36)

1.3.7 Convergence Of q-Exponential Functions

The series of non-negative terms in q-calculus converges if a bounded sequence is formed by

its partial sums, so for two classical q- exponential functions we can find interval of

convergence as follows

Let

eqx=∑

k=0

∞ xk

[k ] !q

Then by using De-Alembert theory

limk → ∞| xk +1

[ k+1 ] !q

xk

[ k ] !q|= lim

k → ∞| xk+1

[ k+1 ] !q∙

[ k ] !q

xk |,

9

¿ limk → ∞| x

[ k+1 ]q|,

using [ k+1 ]q=qk+1

q−1

¿ limk → ∞|x(q−1)

qk+1−1 |,¿|x||q−1|<1 ,

Hence converges and the interval of convergence is

|x|< 1¿1−q∨¿ .¿

Similarly we can prove the other q-exponential function as

Let

Eqx=∑

k=0

∞

qk (k−1 )

2 xk

[ k ] !q

By using De-Alembert we see that

limk → ∞|q( k+1 ) k xk+1

[k+1 ] !q

qk (k−1) xk

[ k ] !q|=|q( k+1) k xk+1

[ k+1 ] !q∙

[ k ] !q

qk (k −1 )xk|,

¿ limk → ∞| q (k +1) k x

qk(k−1) [ k+1 ]q|,¿ lim

k → ∞|qk x (q−1)qk+1−1 |=0 .

1.4 q-Trigonometric Functions

By using the well-known Euler formula in terms of exponential function, we can define the q-

analogues of the two trigonometric functions.

Proposition 1.3.1 (Kac.V, and Cheung.P,2002) The sine and cosine q-analogue function are

given by

10

sin q x=eq

ix−eq−ix

2 i, sinq x=

Eqix−Eq

−ix

2i(1.37)

cosq x=eq

ix+eq−ix

2,cosq x=

Eqix +Eq

−ix

2(1.38)

1.4.1 Properties Of q-Trigonometric Functions

We can see from (1.37) and (1.38) that

cosq x cosq x+sin q x sinq x=1(1.39)

1.4.2 q-Derivative Of q-Trigonometric Functions

The q-derivative of the q-trigonometric function is given by

Dq sinq x=cosq x , Dq sinq x=cosq qx (1.40)

Dq sinq x=−sin q x , Dq cosq x ¿−sin q qx(1.41)

And (1.39), (1.40) and (1.41) are being proved by proposition 1.3.1

1.5 Improved q-Exponential Function

There are two exponential functions that are define by Euler in the previous section, both there

are some properties that are lost, for example

eq (−x )= 1Eq ( x )

,Eq (−x )= 1eq ( x )

Which allows us to defined the improved q-exponential function as

1.5.1 Definition ℇ qz

Let ℇ qz be new q-exponential function, and defined as (Jan L. & Cieśliński, 2011)

ℇ qz=eq

z /2 Eqz /2=∏

k=0

∞ 1+(1−q) z2

qk

1−(1−q) z2

qk(1.42)

11

Where eqz∧Eq

z are the standard q-exponential functions. Classical Cayley transformation

motivated the above definition. (the infinite product representation is valid for |q|<1¿.

1.5.2 Basic Definitions on Improved q-Exponential Function

Definition 1.5.2: If a is any real or complex number, then we defined the following terms as

(a ;q )n=∏j=0

n−1

(1−q j a ) ,n∈N (1.43)

(a ;q )0=1(1.44)

{n }=1+q+…+qn−1

12(1+qn+1)

=[n]

12(1+qn+1)

=2(1−qn+1)

(1−q)(1+qn+1)(1.45 )

Therefore

{n }!={1 } {2 }… {n }=[ n ]q ! 2n

(−1 ;q )n

Definition 1.5.2: Bernoulli number can be demonstrated in term of improved q-Bernoulli

number by the following recurrence relation:

∑k=0

n

(nk )q

(−1, q )n−k

2n−k bk ,q−bn , q={1 ,n=10 , n ≠ 1

(1.46)

Where bk ,q is the Bernoulli number.

Definition 1.5.4: (Wikipedia) If x and y are real or complex parameter, then the

summation by Newton expansion in an ordinary case as

( x+ y )n=∑k=0

n

(nk ) xk yn−k .(1.47)

In the same manner, the following q-addition of the expression is define as (Zhang.Z and

JunWang, 2006)

( x⊕q y )n=∑k =0

n

(nk )q

(−1 ,q )k . (−1 , q )n−k

2n xk yn−k , n=0,1,2 ,… (1.48)

1.5.3 Unification Of q-Exponential Functions

The following statement holds true

ℇ qx .ℇ q

y=ℇq ( x⊕q y )

12

Proof

ℇ qx .ℇ q

y=(∑n=0

∞ xn

[ n ] !q

(−1; q)n

2n )(∑m=0

∞ ym

[ m ] !q

(−1 ;q)m

2m )¿∑

n=0

∞ ∞

(∑k=0

n xk

[ k ] !q

(−1 ;q)k

2kyn−k

[ n−k ] !q

(−1;q)n−k

2n−k )…¿∑

n=0

∞ ∞

(∑k=0

n (−1 ;q)k

2k

(−1;q )n−k

2n−kxk

[k ] !q

yn−k

[n−k ] !q )¿∑

n=0

∞ ∞

(∑k=0

n (−1 ;q)k

2k

(−1;q )n−k

2n−kxk

[k ] !q

yn−k

[n−k ] !q )¿∑

n=0

∞ ∞

(∑k=0

n

(nk )q

(−1; q)k .(−1; q)n−k

2n xk yn−k )¿ℇ q ( x⊕ y )

As required.

1.5.4 Improved q-Trigonometric Functions

We can use the natural way to define the new q-sine and q-cosine functions as

s∈¿q x=ℇq

ix−ℇq−ix

2i¿

C osq x=ℇq

ix+ℇq−ix

2(1.49)

1.6 Bernoulli Numbers

In this work Bernoulli numbers will be defined by the exponential generating function

tet−1

=∑n=0

∞

Bnt n

n!(1.50)

We see that the first Bernoulli number is easy to find, i.e.

B0= limt → 0

tet−1

13

¿ limt →0

1e t , L ' Hospital

¿ 1e0

¿1

B1= limt →0

ddt ( t

et−1 )¿ lim

t →0

e t−1−tet

(e t−1)2 , L' Hospital

¿ limt →0

−t2(et−1)

¿ limt →0

−12 e t

¿−12

1.6.1 Recurrence Formula for Ordinary Bernoulli Numbers

Continuing in this way we will use the tool that we have i.e the ordinary exponential function

in order to derive the recurrence formula for Bernoulli numbers

tet−1

=∑n=0

∞

Bnt n

n!

t=(∑n=0

∞ t n

n! )(∑n=0

∞

Bnt n

n! )−∑n=0

∞

Bnt n

n !,

By using the Cauchy product of two series (Rudin, 1964) i.e

Given the two series ∑ an and ∑ bn we write

Cn=∑ ak bn−k (n=0,1,2 , …) ,

Then ∑ cn is said to be the multiplication of the two series.

Going back to our work we see that

(∑n=0

∞

an)(∑n=0

∞

bk)=∑n=0

∞

(∑k=0

n

an bn−k ) ,

14

we obtain

t=∑n=0

∞

(∑k=0

n

Bkt k

k !t n−k

(n−k )! )−∑n=0

∞

Bnt n

n !,

t=∑n=0

∞

(∑k=0

n

Bk (nk) t n

n! )−∑n=0

∞

Bnt n

n !,

t=∑n=0

∞

(∑k=0

n

(nk )Bk ) tn

n !−∑

n=0

∞

Bnt n

n !.

By comparing the power of t we have

∑k=0

n

(nk )B k−Bn={ 1, for n=10 , for others

(1.51)

which is the recurrence formula for Bernoulli numbers.

1.6.2 Kronecker Delta

Proposition 1.6.1 (Riordan, 1968) If Bn is a Bernoulli number number then,

∑k=0

n 1k+1 (nk)Bn−k=δn , 0 ,

Where δ n ,0 is called Kronecker delta

Proof

Prove by Cauchy product on generating function.

Since Kronecker delta is defined as

δ nm={1 , for n=m0 , for n≠m

Then we can write (1.6.2) as

∑k=0

n

(nk )Bk−Bn=δn , 1 ,

Since

(nn)Bn=Bn

when we assume that n−1=m, i.e

∑k=0

n−1

(nk )Bk=δ n ,1 .

15

we will then have

∑k=0

m

(m+1k )Bk=δm , 0 ,

by opening the summation we have

(m+10 )B0+(m+1

1 )B1+…+(m+1m )Bm=δm ,0

∑k=0

∞

( m+1m−k) Bm−k=δm , 0

Why? Because

(m+1m−k )= (m+1 )!

(m−k ) ! (m+1−m+k )!= (m+1 ) m!

(k+1 )!k ! (m−k )!=(m

k ) m+1k+1

Therefore we have

∑k=0

m

(mk ) m+1k+1

Bm−k=δm, 0

¿∑k=0

m

(mk ) 1k+1

Bm−k=(m+1)−1δm , 0={ 1m+1

, for m=0

0 , for m≠ 0

Which is the same as proposition 1.6.1

∑k=0

n 1k+1 (nk)Bn−k=δn , 0 ,

1.6.3 Lemma Explicit Definition of Bernoulli Number (Arakaya.T & et.al, 2014)

Bernoulli number satisfy the recurrence

∑k=0

n

(n+1k )Bk=n+1 , for n≥ 0(1.52)

1.6.4 Proposition Bernoulli Numbers as Rational numbers (Arakaya.T & et.al, 2014)

The Bernoulli numbers are rational numbers.

Solving for the first seven of the Bernoulli numbers using the above recurrence

B0=1 , B1=−12

, B2=16

, B3=0 ,B4=−130

, B5=0 , B6=1

40, …

16

1.6.5 Bernoulli Polynomials

When we multiply the left hand side of (1.51) with ext and rise it to the power of some

arbitrary constant say α a real or complex parameter.

( te t−1 )

α

ext=∑n=0

∞

Bn(α )(x ) t n

n!(1.53)

It is called the generating function for Bernoulli polynomial.

The generalized Bernoulli polynomials are given as

Bn (x )=∑k=0

n

(nk )Bn−k xk (1.54)

Then by using the above expression we obtain the few Bernoulli polynomials:

B0 (x )=1

B1 ( x )=x−12

B2 ( x )=x2−x+ 16

B3 (x )=x3−32

x2

+ 12

x

B4 ( x )=x4−2 x3+x2− 130

B4 ( x )=x5−52

x4

+ 53

x3−16

x

1.6.6 Some Properties Of Bernoulli Polynomials

Proposition1.6.2 (Zhang.Z and JunWang, 2006) for all integers greater than or equal to one,

then the following holds

Bn (x )=∑k=0

n

(nk )Bn xn−k

Proof (Kac.V and Cheung.P, 2002)

Let

17

Fn ( x )=∑k =0

n

(nk)Bn xn−k

It is quiet simple to see that (a) Fn (0 )=Bn , for n≥ 0. (b) Fn' ( x )=nFn−1 ( x ) , for n≥ 1

Since these characters uniquely characterize Bn(x) you see that the (a) is so simple to find out

because

Fn (0 )=(nk )Bk xn+( n

n−1) Bn−1 xn−1+…+(nn)Bn x0

Fn (0 )=0+0+…+(nn)Bn=Bn

But for (b) and using the fact that if n>k≥ 0 ,

Differentiating both side and for n≥ 1, we see that

ddx

Fn

( x )=∑k=0

n−1

(nk )(n−k )Bn xn−k−1

But

(n−k )(nk)= n !

(n−k−1)

Multiplying and dividing by n we get

ddx

Fn ( x )=∑k=0

n−1

n(nk )B k xn−1−k=n∑k=0

n−1

(nk )B k xn−k−1 ,

as required.

Now from the Bernoulli polynomial generating function we deduced the following

proposition:

Proposition (1.6.3) (Zhang.Z and JunWang, 2006) If α and β are two real or complex

parameters then we say that

Bn(α +β ) (x+ y )=∑

k=0

n

(nk )Bn

( α ) ( x ) Bn−k( β ) ( y )(1.55)

Proof

Here we see that there are two polynomial with variables of x∧ y with index α +β multiplying

themselves.

It follows from (1.54)

18

( te t−1 )

α

ext=∑n=0

∞

Bn(α )(x ) t n

n !,

( te t−1 )

β

e yt=∑n=0

∞

Bn( β )( y ) t n

n!,

( te t−1 )

α+ β

e(x+ y)t=(∑n=0

∞

Bn( α )(x) t n

n! )(∑n=0

∞

Bn( β )( y ) t n

n !) ,

¿∑n=0

∞

(∑k=0

n

Bk(α )(x) tk

k ! )(∑n=0

∞

Bn−k( β ) ( y ) tn−k

(n−k )! ) ,

¿∑n=0

∞

(∑k=0

n

(nk )Bn( α )(x)Bn−k

( β ) ( y )) tn

n!,

Therefore we have

∑n=0

∞

Bn( α+β ) ( x+ y ) t n

n !=∑

n=0

∞

(∑k=0

n

(nk )Bn( α )(x )Bn−k

( β ) ( y )) t n

n!,

by comparing the coefficient of t n

n ! ,

we have

Bn(α +β ) (x+ y )=∑

k=0

n

(nk )Bn

( α ) ( x ) Bn−k( β ) ( y ) .

as required.

But

Bn(0 ) ( x )=xn(1.56)

when we interchange x and y in the addition (1.54) and put β=0 the equation yields

Bn(α ) ( x+ y )=∑

k=0

n

(nk )Bn(α ) ( y ) xn−k ,(1.57)

As a special case by putting α=1 we have

Bn (x+ y )=∑k=0

n

(nk )Bn( y )xn−k (1.58)

19

CHAPTER 2

BERNOULLI MATRIX AND SOME PROPERTIES

Majority of the work in this chapter was presented from a Journal Bernoulli matrix and its

algebraic properties (Zhizheng & JunWang, 2006) and any other information was cited by

means of reference.

2.1 Bernoulli Matrix

Before we talk about the Bernoulli matrix, let recall the definition of the algebra matrix that

says:

2.1.1 Definition Ordinary Matrix and some Properties

The rectangular arrangement of numbers is what we referred to as Matrix. A matrix that has m

number of rows and n number of columns is said to be of size m× n and can be displayed as:

A=[ a11 a12 ⋯ a1n

a21 a22 ⋯ a2n

⋮ ⋮ ⋱am1 am 2 ⋯ amn

]=[aij ]

,

where, the entries a ij are real numbers and they can also be complex in some other kind of

matrix.

Matrix has some properties as follows:

Supposes a matrix A=[aij ] and a matrix B=[b ij ] are m× n matrix and α is a scalar with another

matrix C=[c ij ], then,

Matrix addition is defined as: A+B=[aij+bij ]

Matrix subtraction is defined as: A−B=[a ij−bij ]

Scalar multiplication is defined as: αA=[α aij]

20

Multiplication of matrix is defined as: AC=[∑k=1

n

aik cik ],a.nd so many other properties. Now lets go back to the Bernoulli matrix.

2.1.2 Definition Bernoulli Matrix and Bernoulli Polynomials

Supposed Bn is an nth Bernoulli number and Bn (x ) is a Bernoulli polynomial and α∈ R, then

the generalized Bernoulli matrix Bα=[ Bijα ] and Bernoulli polynomial matrix

Bα ( x )= [B i , j(α ) ( x ) ] wherei , j=0,1,2 , …n. are defined respectively as (Zhang.Z and JunWang,

2006)

[B¿¿α ]ij={( ij)B

i− j

α

, if i≥ j

0 , otherwise ,(2.1)¿

[B¿¿i , j ( α ) ( x )]ij={( ij)B

i− j

(α )

( x ) , if i≥ j

0 ,otherwise ,(2.2 )¿

B(1 ) ( x )=B ( x ) is the Bernoulli polynomial matrix, and B (0 )=B is the Bernoulli matrix

Example: the 3×3 Bernoulli polynomial is given by

2.1.3 Theorem Bernoulli Polynomial Matrix of x and y

The following relation for generalized Bernoulli matrix holds true

B(α +β ) (x+ y )=B (α ) ( x ) B( β ) ( y )=B( α ) ( y ) B( β ) ( x )(2.3)

21

[ B (x ) ]3×3=[ 1 0 02 x−1 1 0

3 x2−3 x+ 12

3 x−32

1 ]

Proof

(Zhang.Z and JunWang, 2006)

In general case we need to show that the sum of the two Bernoulli matrices with the index

(α +β ) from the L.H.S is equal to their product at the R.H.S.

Now, the Bernoulli matrix of the L.H.S has the element of the form

[ B(α +β )(x+ y) ]ij={( ij)Bi− j

(α+β ) (x+ y ) , if i≥ j ,

0 , for others .

But on the R.H.S since it has to do with the multiplication of two matrices, therefore the

number of the row of the first Bernoulli matrix has to be the same as the number of column of

the other Bernoulli matrix. So that the elements are

[ B(α )( x)]ik={( ik )B i− j

(α) (x ) , if i≥ k ,

0 , for others .

[ B(β)( y)]kj={(kj)Bi− j

( β) ( y ) ,if k≥ j ,

0 , for others .

Now multiplying those two matrices we see that

( [B(α )(x )] [ B(β )( y) ])ij=∑k=1

n

[ B(α )(x )]ik [ B(β)( y)¿ ]kj .

Opening that summation we get

¿ [ B(α ) ( x ) ]i 1 [ B( β ) ( y ) ]1 j+ [ B( β ) ( y ) ]i2 [ B( β ) ( y ) ]2 j+…+ [B (α ) ( x ) ]ii [B ( β) ( y ) ]ij+[ B( α ) (x ) ]i (i+1) [B ( β) ( y ) ](i+ 1) j+…+ [B( α ) (x ) ]¿ [ B( β ) ( y ) ]nj

On the i index over there up to the point that we are going to reach ii index the result is not

equal to zero but after that where the k index become greater than i index it becomes zero.

These are the zeros of i index, what about the zeros of j index? When ever k< j the result will

be zero up to the point where k= j,

now assume that i≥ j, we have

¿ [ B(α ) ( x ) ]ij [ B( β ) ( y ) ] jj+[ B( β ) ( y ) ]ij+1 [ B( β ) ( y ) ] j+1 , j+…+ [ B( α ) ( x ) ]ii [ B( β ) ( y ) ]ij

¿( ij)Bi− j

α ( x )( jj)B j− j

β ( y )+( ij+1)B i− j−1

α ( x )( j+1j )B1

β ( y )+…+(ii) Bi−iα ( x )( i

j)B i− jβ ( y )

22

All the terms with the i< j becomes zero leaving first and the last term with

( jj)B j− j

β ( y )=(ii) Bi−iα ( x )=1

Now, it follows from (1.6.6), by simplifying the combinatorial

( ij)B i− j

α (x )( jj) B j− j

β ( y )=∑k= j

i

( ik) Bi−k

α (x )(kj) Bk− j

β ( y )

∑k= j

i

( ik )B i−k

α ( x )(kj)Bk− jβ ( y )=∑

k = j

i

( ij)( i− j

k− j)Bi−kα ( x ) Bk− j

β ( y )

¿( ij)∑k=0

i− j

(i− jk )B

i− j−k

α

( x ) Bkβ ( y )

¿( ij)B

i− j

( α+β )

( x+ y ) ,

Which implies (2.3)

2.1.4 Corollary

B( α1+α 2+…+α n ) ( x1+x2+…+ xn)=Bα 1 ( x1 ) Bα2 ( x2 ) …Bαn ( xn ) .(2.4)

Proof Using mathematical induction

And taking x1=x2=…=xn=x ,α 1=α 2=…=α n=α .

we see that for n=1

Bα 1 ( x1 )=Bα1 ( x1) ,

we assume that it is true for n=k,

B( α1+α 2+…+α k ) ( x1+x2+…+xk )=Bα 1 ( x1 ) Bα 2 ( x2 ) …Bαk ( xk ) .

For n=k+1 gives

B( α1 +α 2+…+α k+α k+1 ) ( x1+x2+…+xk+xk+1 )

¿ Bα 1 ( x1 ) Bα2 ( x2 ) …Bαk ( xk ) Bαk +1 ( xk+1 ) .

Therefore the right hand side becomes

B( α1+α 2+…+α k ) ( x1+x2+…+xk ) Bα k+1 ( xk+1 ) ,

which satisfies theorem (2.1).

We can further assume that x=0 or α=1 , then the Bernoulli matrix will have a simple

powers.

2.1.5 Corollary (Zhang.Z and JunWang, 2006)

23

If

(B¿¿α (x))k=B( kα ) (kx ) ,(2.5)¿

and specially,

( B ( x ))k=B(k ) ( kx )

(B¿¿(α ))k=B (kα )¿

Proof:

By applying mathematical induction on α in all the above expression we see that it is true.

2.1.4 Definition Inverse Of Bernoulli Matrix

Let D= [d i , j ] (0 ≤i , j ≤n ) be (n+1 ) × (n+1 ) matrix which is defined as (Zhang.Z and JunWang,

2006)

d i , j={ 1i− j+1 ( i

j)if i ≥ j ,

0 , otherwise .(2.6)

Theorem 2.1.4: Inverse of Bernoulli matrix can be defined by the previous definition of D.

That means

B−1=D .

Also,

(B¿¿(k ))−1=D k¿

Proof

We need to show that D is the inverse of B.

[B]ij={( ij)Bi− j , if i ≥ j ,

0 , otherwise .

Di , j={ 1i− j+1 ( i

j)if i ≥ j ,

0 , otherwise .

Now if we take their matrix we have

[ BD ]i , j={1 ,if i= j0 , if i≠ j

=δ i , j=[ B ] [ D ]=I(n+1 )×(n+1)

which is going to be a of delta kronecker matrix

24

[ BD ]ij=∑k=0

i−1

[B]ik[ D ]kj .

expanding the summation we have

[ [ B ] [ D ] ]i , j=[ B ]i , 0 [ D ]0 ,i+ [ B ]i, 1 [ D ]1 i+… [B ] ii [ D ] ij+…+ [ B ]i ,i+1 [ D ]i+1 j+…+ [ B ]¿−1 [ D ]n−1 j+ [ B ]i , n [ D ]n ,i

Let i= j

[ [ B ] [ D ] ]i , j=[ B ]i , 0 [ D ]0 ,i+ [ B ]i, 1 [ D ]1 i+…+ [ B ]i , n [ D ]n ,i

Treating the two matrices in terms of their component

∑k= j

i

( ik )B i−k

1k− j+1 (k

j )=( ij)∑k= j

i 1k− j+1 ( i− j

k− j )Bi−k ,

¿( ij)∑k=0

i− j 1k+1 (i− j

k )Bi− j−k=( ij)δi− j ,0 ,

Satisfying proposition (1.49)

The result becomes BD=I , i.e., B−1=D . Looking at this result and corollary 2.1.3 we noticed

that

(B¿¿(k ))−1=(B¿¿k )−1=(B ¿¿−1)k=Dk .¿¿¿

as required.(Zhang.Z and JunWang, 2006)

2.2 Bernoulli Matrix and Generalized Pascal Matrix

2.2.1 Definition Pascal Matrix

Supposed x is an unknown variable and n is an integer that is not equal to zero, then the

generalized (n+1)×(n+1) Pascal matrix is denoted as P [ x ]= [ pij ] ,(i , j=0 , 1 , …,n) and

defined as (Zhang.Z and JunWang, 2006)

Pij={ ( ij) x i− j , i≥ j

0 , otherwise .(2.7)

2.2.2 Theorem Relationship Between Bernoulli Polynomial Matrix And Pascal Matrix

For a Bernoulli matrix and Pascal matrix of a non zero real number x

B (x+ y )=P [ x ] B ( y )=P [ y ] B (x )(2.8)

25

Specially (Zhang.Z and JunWang, 2006)

B (x )=P [ x ] B .(2.9)

Proof.

The matrix on the R.H.S in i and j component can be written as

[ B(x+ y) ]ij={( ij) Bi− j ( x+ y ) , if i ≥ j

0 , if i< j

and L.H.S is

[ p(x ) ]ij={( ij)x i− j , if i≥ j

0 , for i< j

[ B( y )]ij={( ij)Bi− j ( y ) , if i≥ j

0 ,if i< j

multiplying the two matrices on the L.H.S

[ P(x )B( y )]ij=∑k=0

i−1

p[ x]ik[B ( y )]kj

expanding the summation

p [ x ]i 0 [B ( y ) ]0 j+ p [ x ]i 1 [B ( y ) ]1 j+…+ p [ x ]ii [B ( y ) ]ij+…

+ p [ x ]i ,i+1 [ B ( y ) ]i+1 j+…+ p [ x ]¿−1 [B ( y ) ]n−1 j+ p [x ]¿ [B ( y ) ]nj

assume i≤ j

p[ x ]ij [B ( y )]jj+ p[ x]i , j+1[B ( y )] j+1 , j+…+ p [ x ]ii [ B ( y ) ]ijBy putting in terms of their respective matrix

( ij) xi− j( i

j)B j− j ( y )+( ij+1) xi− j−1 B1 ( y )+…+( i

i) xi−i( ij)B i− j ( y )

All the terms with i< j component disappear leaving the ones with i≥ j

Since for i= j we have

(ii)x i−i=( jj)B j− j ( y )=1

∑k= j

i

( ij) xi− j( j

j )B j− j ( y )=∑k = j

i

( ik )x i−k (k

j) Bk− j ( y )

26

¿( ij)∑k= j

i

( i− jk− j)Bk− j ( y ) x i−k

¿( ij)∑k=0

i− j

(i− jk )Bk ( y) x i− j−k=( i

j)B i− j(x+ y) ,

Which give B (x+ y )=P(x)B ( y ) .

In the same manner we can obtain the other part of (2.8). i.e.

B (x+ y )=P(x)B ( x )

Proof.

The matrix on the R.H.S in i and j component can be written as

[ B(x+ y) ]ij={( ij) Bi− j ( x+ y ) , if i ≥ j

0 , if i< j

and L.H.S is

[ p[ y ]]ij={( ij) y i− j ,if i ≥ j

0 , for i< j

[ B( y ) ]ij={( ij)Bi− j ( x ) ,if i ≥ j

0 , if i< j

multiplying the two matrices on the L.H.S

[ p [ y ] B (x)]ij=∑k=0

i−1

p [ y ]ik [B ( x )]kj

expanding the summation

p [ y ]i 0 [B ( x ) ]0 j+ p [ y ]i 1 [B ( x ) ]1 j+…+ p [ y ] ii [B ( x ) ]ij+…

+ p [ y ]i , i+1 [B ( x ) ]i+1 j+…+ p [ y ]¿−1 [B ( x ) ]n−1 j+ p [ y ]¿ [ B ( x ) ]nj

assume i≤ j

p[ y ]ij [B ( x )]jj+ p[ y ]i , j+1[B ( x )] j+1 , j+…+ p [ y ]ii [B ( x ) ]ijby putting in terms of their respective matrix

( ij) y i− j( i

j)B j− j ( x )+( ij+1) y i− j−1 B1 ( x )+…+( i

i) y i−i( ij )Bi− j ( x )

all the terms with i< j component disappear leaving the ones with i≥ j

since for i= j we have

27

(ii) y i−i=( jj)B j− j ( x )=1

by considering the terms where i> j we see that by using (1.6.8),

∑k= j

i

( ij) y i− j( j

j)B j− j ( x )=∑k = j

i

( ik ) y i− k(k

j)Bk− j ( x )

¿( ij)∑k= j

i

( i− jk− j)Bk− j ( x ) y i−k

¿( ij)∑k=0

i− j

(i− jk )Bk (x) y i− j−k=( i

j)B i− j (x+ y )

which give B (x+ y )=P [ y ] B ( x ) .In the same manner, we can obtain B (x+ y )=P [ x ] B ( y ) .

Example

B( x )=(1 0 0 0 ⋯x 1 0 0 ⋯

x2−x+ 16

2 x−1 1 0 ⋯

x3−32

x2+12

x 3 x2−3 x+ 12

3 x−32

1 ⋯

⋯ ⋯ ⋯ ⋯ ⋯)(n+1)×(n+1)

=(1 0 0 0 ⋯x 1 0 0 ⋯x2 2 x 1 0 ⋯x3 3 x2 3 x 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)(n+1)×(n+1)(

1 0 0 0 ⋯

−12

1 0 0 ⋯

16

−1 1 0 ⋯

0 12

−32

1 ⋯

⋯ ⋯ ⋯ ⋯ ⋯

)(n+1)×(n+1)

¿ P [ x ] B.

2.2.3 Theorem Inverse Of Bernoulli Polynomial Matrix and Pascal Matrix

B−1 ( x )=B−1 [−x ]=DP [−x ] (2.10)

(Zhang.Z and JunWang, 2006)

Proof:

Lets try and see how it operate before getting into the details of the proof

28

Pascal matrix is given as

[ P[ x] ]ij={( ij)x i− j , if i≥ j

0 , for i< ji.e

P[x]=

(1 0 0 0 ⋯x 1 0 0 ⋯x2 2 x 1 0 ⋯x3 3 x2 3 x 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)and its inverse as

[ P[ x ]−1 ]ij={( ij)(−x )i− j ,if i ≥ j

0 , for i< jI.e

P [ x ]−1

=

(1 0 0 0 ⋯

−x 1 0 0 ⋯x2 −2 x 1 0 ⋯

−x3 3 x2 −3 x 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)multiplying them we have

P[x]P[-x] =

(1 0 0 0 ⋯x 1 0 0 ⋯x2 2 x 1 0 ⋯x3 3 x2 3 x 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)(1 0 0 0 ⋯

−x 1 0 0 ⋯x2 −2 x 1 0 ⋯

−x3 3 x2 −3 x 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)

=(1 0 0 0 ⋯0 1 0 0 ⋯0 0 1 0 ⋯0 0 0 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)=I

There are three cases involved

first

29

assume i< j

∑k=0

n

P [x ]ik P [−x ]kj=P[ x ]i 0 P[−x ]0 j+P [x ]i 1 P [−x ]1 j+…

+P [ x ]ii P [−x ]ij+P [ x ]i ,i+1 P [−x ]i+ 1, j+…+P [x ]¿ P[−x ]nj

¿0+0+0+…+0=0 ,

Second

assume i= j

¿ P[ x ]i 0 P[−x ]0 j+P [x ]i 1 P [−x ]1 j+…

+P [ x ]ii P [−x ]ij+P [x ] i ,i+1 P [−x ]i+ 1, j+…+P [ x ]¿P [−x ]nj

¿0+(ii)(x )i−i( jj)(−x) j− j+0=0+1+0+…=1 ,

The last case,

assume that i> j

∑k= j

i

( ik ) ( x )i−k (k

j)(−x )k− j=∑k= j

i

( ik )(kj) ( x )

i−k

(−1 )k− j ,

changing the boundary of summation we get

¿∑m=0

i− j

( ij)

( i− j ) (i− j−1 ) …(i− j−m)m!

x i− j(−1)m ,

¿( ij)x

i− j

∑m=0

i− j (i− j ) !(i− j−m )!m !

(−1)m,

¿( ij)x

i− j

∑m=0

l l !( l−m )! m!

(−1)m ,

¿( ij)x

i− j

∑m=0

l

( lm)(−1)m .

Now from Newton expansion formula of binomial that says

(a+b )n=∑k=0

n

(nk)ak bn−k=∑

k=0

n

(nk ) (−1 )n−k=0

If a=1 , b=−1 or a=−1 , b=1 , then

the final expression becomes

¿( ij)x

i− j

∑m=0

l

( lm) (1−1 )m=0

30

According to Theorem 2.2.1, we have

B (x+ y )=P [ x ] B ( y ) ,

if y=0, we have

B (x )=P [ x ] B ,

According to the discussion that we have above that

(P [ x ])−1=P [−x ] ,

multiplying from the left side by P[−x ]

P [− x ] B ( x )=B ,

now multiplying by B−1 ( x )

P [− x ]=B B−1 (x ) ,

B−1 ( x )=DP [−x ] .

Since B−1=D

Example

B−1( x )=(1 0 0 0 ⋯x 1 0 0 ⋯

x2−x+ 16

2 x−1 1 0 ⋯

x2− 32

x2+ 12

x 3 x2−3 x+ 12

3 x−32

1 ⋯

⋯ ⋯ ⋯ ⋯ ⋯)(n+1 )×(n+1)

−1

=(1 0 0 0 ⋯12

1 0 0 ⋯

13

1 1 0 ⋯

14

1 32

1 ⋯

⋯ ⋯ ⋯ ⋯ ⋯

)(n+1)×(n+1)

×(1 0 0 0 ⋯

−x 1 0 0 ⋯x2 −2 x 1 0 ⋯

− x3 3 x2 −3 x 1 ⋯⋯ ⋯ ⋯ ⋯ ⋯

)(n+1)×(n+1)

¿ DP [− x].

31

CHAPTER 3

Q-BERNOULLI MATRIX AND ITS PROPERTIES

Majority of the work in this Chapter was presented from a journal ‘Q-Bernoulli Matrices and

Their Some Properties’ (Naim and KUS, 2015) and any other information was cited by means

of reference.

3.1 Q-Bernoulli Matrix

Having studying the basic concept of q-calculus in Chapter 1 and Bernoulli matrix with some

of its properties in Chapter 2 of this theses, we now continue to see how the q-Bernoulli matrix

is being defined by using the q-Bernoulli polynomials and then some other properties of it,

3.1.1 Definition q-Bernoulli numbers

For integer n ≥ 1 and the Bernoulli numbers Bn. The q-Bernoulli numbers Bn (q ) are defined as

(Naim and KUS, 2015)

Bn (q )=Bn[ n ]q!n !

(3.1 )

By using the above definition we see that the first six q-Bernoulli numbers are:

B0 (q )=B0[ 0 ]q!0 !

=1 ,

B1 ( q )=B1[ 1 ]q !1!

=−12

,

B2 ( q )=B2[2 ]q !2!

=[ 2 ]q!12

,

B3 (q )=B3[3 ]q !3 !

=0 ,

B4 (q )=B4[ 4 ]q !4 !

=[ 4 ]q!720

,

B5 (q )=B5[5 ]q !5 !

=0 ,

32

B6 (q )=B6[ 6 ]q!6 !

=[6 ]q!

30240.

3.1.2 Definition q-Bernoulli Polynomials

The q-Bernoulli polynomials Bn (x , q ) as is defined as (Naim and KUS, 2015)

Bn (x , q )=∑k=0

n

(nk )

qB

n(q ) xn−k (3.2)

The first six q-Bernoulli polynomials also can be seen to be:

B0 (x ,q )=1 ,

B1 ( x , q )=x−12

,

B2 ( x , q )=x2−x+[ 2 ]q!12

,

B3 (x , q )=x3−3 x2

2+

[ 2 ]q! x4

,

B4 ( x ,q )=x4−2x3+[2 ]q x2

2+

[ 4 ]q!720

,

B5 (x , q )=x5−5 x4

2+

5 [ 2 ]q x3

6+

[4 ]q ! x144

.

Theorem 3.1.3 (Naim and KUS, 2015) When we apply the commutativity property on x and

y such that xy=qyx, then

Bn (x+ y , q )=∑k=0

n

(nk )

qyn−k Bk ( x , q ) ,(3.3)

with the same approached on the theorem we get

Bn (x+ y , q )=∑k=0

n

(nk )

qxn−k Bk ( y ,q )(3.4)

3.1.3 Definition q-Bernoulli Matrix

If Bn (q ) is an nth q-Bernoulli number, then the (n+1 ) × (n+1 ) q-Bernoulli matrix Bij (q )=[b ij (q ) ] is defined as (Naim and KUS, 2015)

33

Bij (q )={( ij)q

b i− j (q ) , if i≥ j

0 , otherwise(3.5)

where 0≤ i, j ≤ n .

For example

B(q )=(1 0 0 0 0

−12

1 0 0 0

[2 ]q2⋅3 !

[ 2 ]q2! 1 0 0

0[ 3 ]q [2 ]q

2⋅3 ![ 3 ]q2 ! 1 0

[ 4 ]q30⋅4 !

0[ 4 ]q [ 3 ]q

2⋅3 ![ 4 ]q2!

1)

3.1.4 Definition q-Bernoulli Polynomials Matrix

Suppose Bn (x , q ) is a q-Bernoulli polynomial. The q-Bernoulli polynomial matrix as (Naim

and KUS, 2015)

Bij ( x , q )={( ij)q

Bi− j ( x , q ) , if i≥ j

0 , otherwise(3.6)

3.1.5 Theorem Inverse of q-Bernoulli Matrix

Let D (q )=[d ij (q ) ] be (n+1 ) × (n+1 ) matrix that is defined as (Naim and KUS, 2015)

d ij ( q )={(nk )q

[ i− j ]q !(i− j+1 )!

, if i≥ j

0 , otherwise(3.7 )

Then D (q ) is called the inverse of the q-Bernoulli matrix.

Proof (Naim and KUS, 2015) If B (q ) is the q-Bernoulli matrix, and using the definition of

D (q ) above, then

34

to prove this there is need to show that the multiplication [ B (q ) D(q)] is equal to q-kronecker

delta.

Now

[ B (q ) ]ik={( ik)q

Bi−k (q ) , if i≥ k

0 , otherwise

[ D (q ) ]kj={(kj)q

[ k− j ]q!(k− j+1 )!

, if k ≥ j

0 , otherwise

Putting them in their matrix form [ B (q ) D(q)]ij=∑k=0

n

[B (q ) ]ik [ D (q ) ]kj

expanding the summation we have

[ B (q ) ]i 0 [D (q ) ]0 j+[ B (q ) ]i 1 [D (q ) ]1 j+…+[ B (q ) ]¿ [ D (q ) ]nj ,

assume that i≤ j

[ B (q ) ]ij [ D (q ) ] jj+ [B (q ) ]i , j +1 [ D (q ) ] j+1 , j+…+ [B (q ) ]ii [D (q ) ]ij ,

( ij)q

Bi− j (q )( jj)q

[ j− j ]q!( j− j+1 ) !

+( ij+1)

qB i− j−1 (q )( j+1

j )q

[1 ]q !(2 ) !

+…+(ii)qBi−i (q )( i

j)q

[i− j ]q !( i− j+1 ) !

,

All the component with i< j becomes zero leaving the terms with

(ii)q=( j

j)q=1

Then by considering the terms which are not equal to zero and one

[ B (q ) D(q)]ij=∑k=0

n

Bik D kj(q) ,

¿∑k= j

i

( ik )

qBi− k(q)(kj)q

[ k− j ]q !(k− j+1 )!

,

¿∑k= j

i

( ik )

q(kj)q

[ k− j ]q!(k− j+1 )!

Bi−k (q ) ,

¿( ij)q∑k= j

i

( i− jk− j)q

[k− j ]q !( k− j+1 )!

bi−k (q ) ,

Let t=k− j

35

¿( ij)q

∑k= j

i

(i− jt )

q

[t ]q !( t+1 ) !

bi−k (q ) ,

Shifting the summation to start from t=0 to i− j

¿( ij)q∑t=0

i− j

(i− jt )

q

[t ]q !( t+1 ) !

bi− j−t (q ) ,

and multiplying by [i− j−t ]q !(i− j−t )!

¿( ij)q∑t= j

i− j

(i− jt )

q

[t ]q !( t+1 ) !

Bi− j−t[ i− j−t ]q !( i− j−t ) !

,

¿( ij)q

[ i− j ]q !(i− j )! ∑t= j

i− j

(i− jt )

q

1(t+1 ) !

Bi− j−t ,

By using the orthogonality relation for Bernoulli numbers i.e Proposition 1.6.1

gives

[ B (q ) D (q 0 ) ]ij=( ij )q

[ i− j ]q !(i− j ) !

δi− j ,0=δi , j .(3.8)

And (3.8) is of Kronecker delta.

3.2 q-Bernoulli Matrix and q-Pascal Matrices

3.2.1 Definition Pascal Matrix and Inverse of Pascal Matrix

The generalized (n+1 ) × (n+1 ) q-Pascal matrix P ( x , q )=[ pij (q)] is defined as (Naim and KUS,

2015)

pij ( q )={( ij)q

x i− j , if i≥ j

0 , otherwise(3.9)

and the inverse of the generalized q-Pascal matrix P−1 ( x , q ) [ pij' (q ) ] as

Pij' (q )={( i

j)qq(i− j

2 )(−x )i− j ,if i ≥ j

0 , otherwise .

(3.10)

Now, the factorization of q-Pascal matrix can be generalized by the following theorem

36

3.2.2 Theorem Relationship Between q-Bernoulli Polynomial Matrix and Pascal Matrix

(Naim and KUS, 2015) Supposed the q-Bernoulli polynomial matrix B (x , q ) and the

generalized q-Pascal matrix P ( x , q ), then

B (x+ y , q )=P ( y , q ) B ( x , q )=P ( x , q ) B ( y , q )(3.11)

the interchanging occur as a result of commutative property

And specially

B (x , q )=P (x , q ) B (q )(3.12)

Proof (Naim and KUS, 2015) Consider B (x , q ) as the q-Bernoulli polynomial matrix and

P ( y ,q ) as generalized q-Pascal matrix. Then we see that

( P ( y , q ) B ( x ,q ) )ij=∑k=0

n

Pik (q ) Bkj ( x ,q )=∑k=0

n

[P ( y , q )]ik [B ( x , q )]kj ,

Expanding

[ P ( y , q ) ]i 0 [ B ( x , q ) ]0 j+[ P ( y , q ) ]i 1 [B ( x ,q ) ]1 j+…+ [P ( y , q ) ]¿ [ B ( x , q ) ]nj ,

Supposed i≤ j

[ P ( y , q ) ]ij [B ( x ,q ) ] jj+[ P ( y , q ) ]i , j+1 [B ( x ,q ) ] j+1 , j+…+[ P ( y ,q ) ]ii [B ( x ,q ) ]ij ,by expressing them in their matrix form:

( ij)q

y i− j( jj)q

B j− j (x ,q )+( ij+1)

qy i− j−1( j+1

j )qB1 ( x ,q )+…+(ii)q

y i−i( ij)q

B i− j (x ,q ) ,

(ii)q=( j

j)q=1 ,

and

i< j →0

now

( P ( y ,q ) B ( x ,q ) )ij=∑k=0

n

Pik (q ) B kj ( x , q ) ,

¿∑k= j

i

( ik )

qy

i−k

(kj)qB

k− j( x ,q ) ,

¿∑k= j

i

( ij)q

( i− jk− j)q

yi−k

Bk− j ( x ,q ) ,

let t=k− j

37

¿∑k= j

i

( ij)q

(i− jt )

qy

i−k

Bt (x , q ) ,

shifting the summation to start from t=0¿ i− j

¿( ij)q

∑t=0

i− j

(i− jt )

qy

i− j−t

Bt ( x , q ) ,

comparing it with (3.3 ), we have the equivalent

( P ( y ,q ) B ( x , q ) )ij=( ij)q

Bi− j

( x+ y ,q )= (B ( x+ y , q ) )ij ,

Similarly we can obtain the second part of (3.2.3). i.e.

B (x+ y , q )=P (x ,q ) B ( y ,q ) .

Proof (Naim and KUS, 2015)

( P ( x ,q ) B ( y , q ) )ij=∑k=0

n

Pik (x , q ) Bkj ( y ,q )=∑k=0

n

[ P (x ,q )]ik[B ( y , q )]kj

Opening the summation

[ P ( x ,q ) ]i 0 [B ( y , q ) ]0 j+[ P ( x ,q ) ]i 1 [B ( y , q ) ]1 j+…+ [P ( x , q ) ]¿ [B ( y ,q ) ]nj

Supposed i≤ j

[ P ( x , q ) ]ij [B ( y ,q ) ] jj+ [ P ( x , q ) ]i , j+1 [ B ( y ,q ) ] j+1 , j+…+ [ P ( x , q ) ]ii [ B ( y ,q ) ]ijExpressing them in their matrix form:

( ij)q

x i− j( jj)q

B j− j ( y , q )+( ij+1)

qxi− j−1( j+1

j )qB1 ( y ,q )+…+(ii)q

x i−i( ij)q

Bi− j ( y , q )

(ii)q=( j

j)q=1

and

i< j →0

Now

( P ( x , q ) B ( y ,q ) )ij=∑k=0

n

Pik (x , q ) Bkj ( y , q ) ,

¿∑k= j

i

( ik )

qx

i−k

(kj )q

Bk− j

( y ,q ) ,

¿∑k= j

i

( ij)q

( i− jk− j)q

xi−k

Bk− j ( y ,q ) ,

Let t=k− j

38

¿∑k= j

i

( ij)q

(i− jt )

qx

i−k

Bt ( y , q ) ,

Shifting the summation to start from t=0¿ i− j

¿( ij)q

∑t=0

i− j

(i− jt )

qx

i− j−t

Bt ( y , q ) ,

Comparing it with (3.3 ), we have the equivalent

( P ( x , q ) B ( y , q ) )ij=( ij)q

Bi− j

( x+ y ,q )= (B ( x+ y , q ) )ij .

and also poof (3.12) in similar way with (3.11). i.e.

B (x , q )=P (x , q ) B (q ) .

Proof. (Naim and KUS, 2015)

[ P ( x ,q ) B(q) ]ij=∑k=0

n

[P ( x , q ) ]ik [ B (q ) ]kj ,

expanding the summation we have

[ P ( x , q ) ]i 0 [B ( q ) ]0 j+[ P ( x , q ) ]i 1 [B ( q ) ]1 j+…+ [P ( x ,q ) ]¿ [B ( q ) ]nj ,

assume that i≤ j

[ P ( x ,q ) ]ij [B (q ) ] jj+[ P ( x , q ) ]i , j+1 [B (q ) ] j+1 , j+…+[ P ( x ,q ) ]ii [ B (q ) ]ij ,writing them in their matrix form

( ij)q

x i− j( jj)q

B j− j (q )+( ij+1)q

x i− j−1( j+1j )

qB1 (q )+…+(ii)q

x i−i( ij)q

Bi− j (q ) ,

all the terms with the combination where i< j will tends to zero leaving

(ii)q=( j

j)q=1 ,

then the other part will be

( P ( x , q ) B (q ) )ij=∑k=0

n

Pik (q ) Bkj (q ) ,

¿∑k= j

i

( ik )

qx

i−k

(kj )q

Bk− j

( q ) ,

¿∑k= j

i

( ik )

q(kj)q

xi−k

Bk− j ( q ) ,

¿( ij)q∑k= j

i

( i− jk− j)q

xi−k

Bk− j (q ) ,

39

¿( ij )q

∑k= j

i

(i− jk )

qx

i−k

Bk (q ) ,

Shifting the summation to start from k=0¿ i− j

¿( ij)q∑k=0

i− j

(i− jk )

qx

i−k

Bi− j−k ( q ) ,

¿( ij)q

Bi− j

( x ,q ) ,

¿ ( B ( x , q ) )ij .

Example

3×3 q-Bernoulli polynomial matrix

( P ( y , q ) B ( x , q ) )ij

=

( 1 0 0y 1 0y2 [ 2]q y 1 )×(

1 0 0

x−12 1 0

x2−[ 2 ]q

2x+

[2 ]q12

[2 ]q−[2 ]q

21 )

=(1 0 0

(x+ y )−12 1 0

x2+[2 ]q xy+ y2−[2 ]q

2( x+ y )+

[2 ]q12

[2 ]q ( x+ y )−[2 ]q

21 )

¿ B (x+ y ,q )

( P ( x ,q ) B (q ) )ij=¿

(1 0 0 0x 1 0 0x2 [2 ]q x 1 0x3 [3 ]q x2 [3 ]q x 1 )×(

1 0 0 0

−12

1 0 0

[2 ]q2⋅3 !

[2 ]q2 !

1 0

0[2 ]q [ 3 ]q

2⋅3 ![ 3 ]q2 !

1)

40

=(1 0 0 0

x−12

1 0 0

x2−[2 ]q

2x+

[2 ]q12

[ 2 ]q x−[2 ]q

21 0

x3−[3 ]q

2x2+

[2 ]q [ 3 ]q12

x [ 3 ]q x2−[2 ]q [ 3 ]q

2+[ 2 ]q [3 ]q

12[ 3 ]q x−

[3 ]q2

1)

¿ B (x , q )

3.2.3 Definition Inverse of q-Bernoulli Polynomials Matrix

(Naim and KUS, 2015) If B (x , q ) is a q-Bernoulli polynomial matrix, then B−1 ( x ,q )=[c ij (q ) ], where

c ij ( q )={ [ i ]q ![ j ]q !∑t=0

i− j q( t2)(− x)t

[ t ]q! (i− j−t +1 )!, if i≥ j

0 , otherwise .

,(3.13)

3.2.4 Corollary

(Naim and KUS, 2015) Let P ( x , q ) be the generalized q-Pascal matrix and B (q ) be q-Bernoulli matrix. Then we can use the factorization of B (x , q ) in (3.12)

B−1 ( x , q )=B−1 (q ) P−1 ( x ,q )=D ( q ) P−1 ( x , q ) .

With the inverse of generalized q-Pascal matrix (3.10), and considering the R.H.S

[ D (q ) ]ik={( ik )

q

[ i−k ]q !( i−k+1 )!

, if i≥ k ,

0 , otherwise .

Pkj' (q )={(k

j)qq(k− j

2 )(−x )k− j , if k ≥ j ,

0 , otherwise .

[ D (q ) P−1(x , q)]ij=∑k=0

n

[ D (q ) ]ik [ P−1(x ,q)]kj ,

¿ [ D (q ) ]i 0 [ P−1(x ,q) ]0 j+[ D (q ) ]i 1 [ P−1(x , q)]1 j+…+ [D (q ) ]¿ [ P−1 (x ,q ) ]nj ,

assume that i≤ j

41

¿ [ D (q ) ]ij [P−1(x ,q)] jj+[ D (q ) ]i , j+1 [ P−1(x , q)] j+1 , j+…+[ D (q ) ]ii [ P−1 ( x , q ) ]ij ,

¿( ij )q

[ i− j ]q !( i− j+1 )! ( j

j)qq

( j − j2 ) (−x ) j− j+( i

j+1)q

[ i− j−1 ]q !(i− j )! ( j+1

j )qq

(12 )(−x )1+¿

…+[ i−i ]q!

(1 )! ( ij)q

q(i− j

2 ) (−x )i − j ,

but for

(ii)q=( j

j)q=1 ,∧i< j=0

considering the other part

(D(q)P−1( x , q))ij=∑k =0

n

Dik(q)Pkj' (q) ,

¿∑k= j

i

( ik )

q

[ i−k ]q !(i−k+1 )! (k

j )q(−x)

k− j

q(k− j

2 ),

¿∑k= j

i

( ik )

q(kj)q

[ i−k ]q! q(k− j

2 )

( i−k+1 )!(−x )k− j ,

¿∑k= j

i

( ij)q

( i− jk− j)q

[i−k ]q !q(k− j

2 )

( i−k+1 ) !(−x)k− j ,

¿( ij)q∑k= j

i

( i− jk− j)q

[i−k ]q !q(k− j

2 )

( i−k+1 ) !(−x)k− j ,

Let t=k− j

¿( ij)q∑k= j

i

(i− jt )

q

[ i−k ]q! q(t2 )

( i−t +1 ) !(−x )t ,

Shifting the summation to start at t=0¿ i− j

¿( ij)q∑t=0

i− j

(i− jt )

q

[ i− j−t ]q! q(t2 )

( i− j−t+1 ) !(−x )t ,

¿[ i ]q ![ j ]q !∑t=0

i− j q( t2)(−x )t

[ t ]q! ( i− j−t +1 )!,

¿c ij (q ) .

42

CHAPTER 4

IMPROVED Q-BERNOULLI MATRIX AND ITS PROPERTIES

In this chapter, together with the knowledge that we obtained in the previous Books and

Journals, we used some properties in other related materials as they are been cited by means of

reference and at the end we developed another form of Bernoulli matrix called ‘ THE

IMPROVED BERNOULLI MATRIX’ where the Bernoulli polynomials are generated with

the improved q-exponential function. But before then lets us give a brief history of q-Bernoulli

numbers.

4.1 History of q-Bernoulli Numbers

Carlitz was the first person to studied the q-analogue of Bernoulli numbers together with

Bernoulli polynomials in the middle of last century where he introduced a new sequence as

{βn }n≥ 0 , and relationship between Bernoulli polynomials and Euler polynomials are been

proved in (H. M. Srivastava & Pint´er, 2004) . and they also presented the generalized

polynomials. Properties of Genocchi polynomials and Euler polynomials are been investigated

by kim et al. in (T., 2006)- (Kim, 2007), some recurrence relationship are also given there, the

q-extension of Genocchi numbers are presented in different manner in (Cenkci & et.al, q-

extensions of Genocchi numbers, 2006) by Cenkci et al. The new concept of the q-Genocchi

number and polynomials are presented by Kim in (Kim, 2007). In (Cenkci & et.al, q-

extensions of Genocchi numbers, 2006), The q-Genocchi zeta function and l−¿ function

through the use of generating functions and Mellin transformation are been discuss by Simsek

et al. in (Simsek & et.al, 2008), There so many recent interesting research on this related area

by so many authors as in: Kurt V. (Kurt V. , 2014), Gabuarry and Kurt B. (Gabaury & Kur,

2012) , Kurt in (Kurt & et.al, 2013), Srivastava in (Srivastava & et.al, 2004), (Srivastava &

Vignat, 2012), Choi in (Choi & et.al, 2008), Nalci and Pashaev in (Nalci & Pashaev, 2012),

Luo in (Luo, 2010), and Srivastava in (Srivastava & Luo, 2006), (Srivastava & Luo, 2011),

(Cenkci & et.al, 2008), and Cheon in (Cheon, 2003).

43

4.1.1 Definition Carlitz q-Bernoulli Number

We first present here the initial recurrence q-Bernoulli number by Carlitz as:

∑k=0

n

(nk ) βk qk+1−βn={1 , n=10 , n>1

From the Bernoulli generating function. i.e

tet−1

=∑n=0

∞

Bnt n

n!

There are a lot definition of the quantum form of Bernoulli number and the Bernoulli

polynomials, we find their differences according to their application, for example we can

defined it by generating function, so because we have a several types of quantum exponential

function, so we have the several types of the q-Bernoulli numbers as well, we can also defined

it arbitrary like its been done in the previous chapter.

4.2 Improved q-Bernoulli Numbers

In this chapter, the classical definition of quantum calculus concept will be used, by recalling

(1.41) of Definition 1.4.2 we can have the following lemma.

4.2.1 Lemma Recurrence Formula For Improved q-Bernoulli Number

We can equivalently define (1.41) by means of the generating function as:

xεq ( x )−1

=∑n=0

∞

bn ,qxn

[n]q !(4.1)

Proof

Let

xεq ( x )−1

=∑n=0

∞

bn ,qxn

[n]q !

Cross and multiply

x=[ εq ( x )−1 ]∑n=0

∞

bn , qxn

[ n ]q !

44

¿∑n=0

∞

bn ,qxn

[n]q !. εq ( x )−∑

n=0

∞

bn ,qxn

[n ]q !

¿∑n=0

∞

bn ,qxn

[n]q !.

(−1 , q )n2n

xn

[n]q!−∑

n=0

∞

bn ,qxn

[n ]q !

By using the Cauchy product of series we have

¿∑n=0

∞

(∑k=0

n

bk ,qxk

[k ]q !.

(−1 , q )n−k

2n−kxn−k

[n−k ]q! )−∑n=0

∞

bn , qxn

[n ]q !

¿∑n=0

∞

(∑k=0

n

[nk ]q bk ,q(−1 , q )n−k

2n−k ) xn

[n]q!−∑

n=0

∞

bn ,qxn

[n ]q !

¿∑n=0

∞

(∑k=0

n

[nk ]q(−1, q )n−k

2n−k . bk , q−bn ,q) xn

[n]q !

By comparing the power of x, we get

¿∑k=0

n

[nk ]q

(−1 ,q )n−k

2n−k . bk ,q−bn ,q={1, n=10 , n≠1

¿∑k=0

n−1

[nk ]q

(−1 ,q )n−k

2n−k bk ,q={1 , n=10 , n≠ 1

(4.2)

Which is the recurrence formula for the improved q-Bernoulli number as required.

By using the expression in (4.2) we can have the first few improved q-Bernoulli numbers as:

b0 ,q=1

b1, q=−12

=−1{2 }q

b2, q=14

q(q+1)q2+q+1

=q [2]q

4 [3 ]q

b3 ,q=0

4.2.3 Lemma Advantage Of Improved q-Exponential Function

All the coefficient of the improved quantum Bernoulli numbers are zero except the initial one.

i.e

bn ,q=0 , for n=2 p+1 , ( p∈ N )

Proof

Let

45

f ( x )=∑n=0

∞

bn , qxn

[ n ]q !

Subtracting the first term in the above expression

¿∑n=0

∞

bn ,qxn

[n ]q!−b1 , q x

But b1, q=−12

¿ xεq ( x )−1

+ x2

¿x2 ( 2

εq ( x )−1+1)

¿ x2 ( ε q ( x )+1

εq ( x )−1 )We assume that f ( x ) is an even function, i.e f ( x )=f (−x ) .

Now,

f (−x )=−x2 ( εq ( x )+1

εq ( x )−1 )From the definition of ε q (x )=Eq( x

2 )eq ( x2 ) implies that

ε q (−x )=Eq(−x2 )eq(−x

2 )=eq−1(−x

2 )Eq−1(−x

2 )=εq−1 (x )

Therefore,

f (−x )=−x2 (

1εq ( x )

+1

1ε q (x )

−1 )Multiplying by

εq ( x )εq ( x ) , we have

¿− x2 (

1εq ( x )

+1

1εq (x )

−1 ) εq ( x )εq ( x )

46

¿− x2 (

εq ( x )εq ( x )

+ε q ( x )

εq (x )εq (x )

−εq ( x ) )¿− x

2 ( 1+εq ( x )1−ε q ( x ) )

¿ x2 ( ε q ( x )+1

εq ( x )−1 )=f ( x )

As required.

And the previous lemma is one of the advantages of the improved q-exponential function over

the ordinary q-exponential function.

4.2.4 Improved q-Bernoulli Polynomials

We can also use the means of generating function to defined the improved q-Bernoulli

polynomials Bn ,qα ( x ) as

( tεq ( t )−1 )

α

ε q ( xt )=∑n=0

∞

bn , qα (x) t n

[ n ]q !(4.3)

Where α is a real or complex parameter.

We can observed that, Bn(1) ( x )=Bn (x ) and Bn

(1) (0 )=Bn are the classical improved Bernoulli

polynomials and classical Bernoulli numbers, respectively.

The Bernoulli polynomials can also be defined with respect to x and y as

tεq ( t )−1

∙ εq ( xt ) ∙ ε q ( yt )=∑n=0

∞

Bn ,q ( x , y ) t n

[n ]q !(4.4)

4.2.5 Theorem Additive Theory

supposed x , y∈C , then

Bn ,q ( x , y )=∑k=0

n

[nk ]qbk, q(x⊕q y)

n−k

=∑k=0

n

[nk ]q

(−1, q )n−k

2n−k bk ,q ( x ) yn−k (4.5)

Proof

By using (4.4)

47

tεq ( t )−1

∙ εq ( xt ) ∙ ε q ( yt )=∑n=0

∞

Bn ,q ( x , y ) t n

[n ]q !

(∑n=0

∞

Bn , qt n

[ n ]q ! )(∑n=0

∞

( x⊕q y )n t n

[ n ]q ! )=∑n=0

∞

Bn ,q ( x , y ) t n

[ n ]q!

∑n=0

∞

(∑k=0

n

Bk , qt k

[ k ]q ! ( x⊕q y )n−k t n−k

[ n−k ]q! )=∑n=0

∞

Bn ,q ( x , y ) t n

[ n ]q !

∑n=0

∞

(∑k=0

n

Bk , q ( x⊕q y )n−k tk

[ k ]q !t n−k

[ n−k ]q! )=∑n=0

∞

Bn ,q ( x , y ) t n

[ n ]q !

∑n=0

∞

(∑k=0

n

[nk ]q Bk , q

( x⊕q y )n−k) t n

[ n ]q !=∑

n=0

∞

Bn , q ( x , y ) t n

[ n ]q !

Comparing the power t we have

∑k=0

n

[nk ]qB

k , q( x⊕q y )

n−k

=∑n=0

∞

Bn,q ( x , y )

As required. For the second equation we must use 4.3.2, then it follows

tεq ( t )−1

∙ εq ( xt )∙ ε q ( yt )=(∑n=0

∞

Bn , q (x ) t n

[n ]q ! )(∑n=0

∞ t n yn

[ n ]q !(−1 , q )n

2n )By using Cauchy product we lead to

(∑n=0

∞

∑k=0

n

(nk)q

Bk , q (x ) yn−k (−1 ,q )n−k

2n−k ) tn

[ n ]q !=∑

n=0

∞

Bn , q (x , y ) tn

[n ]q !

Thus the last equation holds true.

4.2.6 Theorem for a real or complex parameter α , the following holds is true

Bn ,q(α+β )( x⊕q y )=∑

n=0

∞

[nk ]qBn ,q

(α ) ( x ) Bn−k , q( β ) ( y )(4.6)

Proof

Considering the right hand side

( tεq ( t )−1 )

α

ε q ( xt )=∑n=0

∞

bn , qα (x) t n

[ n ]q !

48

( tεq ( t )−1 )

β

ε q ( yt )=∑n=0

∞

bn , qβ ( y ) t n

[ n ]q!

Multiplying them we obtain

( tεq ( t )−1 )

( α+β )

ε q ( x⊕q y ) (t )=(∑n=0

∞

bn , qα (x ) t n

[n ]q ! )(∑n=0

∞

bn ,qβ ( y ) t n

[ n ]q ! )By using the Cauchy product of series formula

¿∑n=0

∞

(∑k=0

n

bk ,qα (x ) t k

[ k ]q !bn−k , q

β ( y) tn−k

[ n−k ]q ! )¿∑

n=0

∞

(∑k=0

n

bk ,qα (x )bn−k ,q

β ( y) t k

[ k ]q!tn−k

[ n−k ]q ! )¿∑

n=0

∞

(∑k=0

n

[nk ]q bk ,qα (x )bn−k ,q

β ( y) t n

[n ]q ! )¿∑

n=0

∞

(∑k=0

n

[nk ]q bk ,qα (x )bn−k ,q

β ( y)) tn

[ n ]q !

But the left hand side is

∑n=0

∞

Bn , q(α+β) ( x⊕q y ) tn

[ n ]q !

By comparing the power of t from the both side

Bn ,q(α+β )( x⊕q y )=∑

k=0

n

[nk ]qbk , q

α (x)bn−k ,qβ ( y )

As required base on the consequence of lemma 4.3 which is equivalent to (1.55)

lemma 4.2.7 The improved Bernoulli polynomials can also be demonstrated as

Bn ,q ( x )=∑k =0

n

[nk ]qbk ,q xn−k (−1 ;q )n−k

2n−k (4.7)

Proof. Put x=1 at additive theorem 4.3.1, then we lead to this equality.

By using the above expression we obtain the first few Bernoulli polynomials as

B0 ,q ( x )=1

B1 ,q ( x )=[2]q x

2−1

2

49

B2 ,q ( x )=[ 2 ]q x2

2 −x+q [2 ]q4 [ 3 ]q

B3 ,q ( x )=[ 2 ]q [ 3 ]q x3

4 −3 [ 2 ]q x2

4 +3 q [ 2 ]q x

4 [3 ]q

4.3 Improved q-Bernoulli Matrix And Its Properties

4.3.1 Definition Improved q-Bernoulli Matrix and Improved q-Bernoulli Polynomials Matrix

The generalized (n+1 ) × (n+1 ) q-improved Bernoulli polynomial matrix as

Bqα ( x )= [B i , j

α (q , x ) ] (i , j=0,1, …, n ) is defined by the following formular

Bi , jα (q , x )={( i

j)q

(−1; q)i− j

2i− j b i− j ,qα ( x ) ,if i ≥ j

0 , otherwise(4.8)

And B(1 ) ( x )=Bq ( x )∧B (0 )=Bq are called the improved q-Bernoulli polynomial matrix and

improved Bernoulli matrix respectively.

Example:

[ Bq( x )]=[ 1 0 02[2 ]q x−2 1 0

[ 2]q x2−2 x+q[ 2 ]q2[3 ]q

[2 ]q x−1 1 ] And

[ Bq ]=[1 0 0

−12 1 0

q[ 2]q

4[ 3 ]q−1 1 ]

Are the example of improved q-Bernoulli polynomial and Bernoulli matrix respectively. When

we tend q to 1 from the left side, we reach to the form of ordinary Bernoulli matrix.

4.3.2 Theorem Improved q-Bernoulli Polynomial Matrix in terms of x and y

The following relation for improved q-Bernoulli matrix with respect to x and y holds true:

¿

Proof

50

The case for i< j , i= j follows the same way as the previous proof in the generalized Bernoulli

matrix and for i> j we have:

∑k= j

i

( ik )

q

(−1 ;q)i−k

2i−k bi−k ,q

( x )(kj)q

(−1; q)k− j

2k− j b i−k ,q ( y )

¿( ij)q∑k=0

i− j

(i− jk )

q

(−1 ;q )i−k− j (−1 ;q )k2i− j b

i−k , q

(x ) bk , q ( y )

¿( ij)q

Bi− j

( x⊕q y )

And the last expression is additive theory as required.

Lemma 4.3.3 The following relation for improved q-Bernoulli numbers hold true

∑k=0

m

(mk )q

(−1;q )m−k

2m−k ∙bm−k

[k+1 ]=δ m,0(4.10)

Proof. At lemma (1.50) we proved this relation for Bernoulli numbers. The proof is exactly

similar to the ordinary case.

Definition 4.3.4 the inverse of improved q-Bernoulli matrix is defined by a matrix D, which is

d i , j(q)={ 1[ i− j+1 ]q ( i

j)qif i≥ j ,

0 , otherwise .(4.11)

4.3.3 Theorem Inverse of Improved q-Bernoulli matrix

Inverse of Improves q-Bernoulli matrix can be defined by the previous definition of D. That

means

B−1 (q )=D ( q ) .(4.12)

Proof

Since both of the matrices are lower triangle, their multiplications are also lower triangle. For

the another entries we may use the similar calculation as follow

¿∑k= j

i

( ik )

q(kj)q

1[ k− j+1 ]q

(−1 ;q )i−k

2i−k bi−k , q

¿( ij)q∑k= j

i

( i− jk− j)q

1[ k− j+1 ]q

(−1 ;q )i−k

2i−k bi−k , q

51

¿( ij )q

∑k=0

i− j

(i− jk )

q

1[ k+1 ]q

(−1; q )i− j−k

2i− j−k bi− j−k ,q=( ij)q

δ i− j , 0

As required.

Satisfying theorem 4.3.3.

52

CHAPTER 5

SUMMARY AND CONCLUSION

There are several definition for q-Bernoulli matrix and we works on a classes of improved q-

Bernoulli matrix which is more suitable to make a q-analogue of the same concept. Some

properties of improved q-Bernoulli numbers and polynomials allowed us to work this concept

easily. At the end of the day, we may defined these numbers by using different generating

functions. But the improved one works better and convinced the ordinary case better.

53

REFERENCE

Cenkci.M, Can.M. & Kurt.V. (2006). q-extensions of Genocchi numbers. J. Korean Maths. J.

Korean Math. Soc. 43(1), 183-198.

Cenkci.M, Kurt.V, Rim.S.H. & Y.Simsek. (2008). On (i; q) Bernoulli and Euler numbers.

Elsivier. 128(2), 280-312.

Cheon, G. (2003). A note on the Bernoulli and Euler polynomials. Elsevier. 16(3), 365-368

Choi. J, Anderson.P. & Srivastava.H. (2008). Some q-extensions of the Apostol-Bernoulli and

Apostol-Euler polynomials of order n and the multiple Hurwitz zeta function. Elsevier.

199(2), 723–737

Gabaury, S. & Kurt. B. (2012). Some relations involving Hermite-based Apostol-Genocchi

polynomials. Apply Mathematics Science. 6(82), 4091-4102.

Srivastava,H.M. & Pinter, A. (2004). Remarks on some relationships between the Bernoulli

and Euler polynomials. Elsevier. 17(4). 375-380.

Kim, T. (2007). A note on the q-Genocchi numbers and polynomials. Journal of Inequality

and application. 2007(71452)8.

Kim.D, Kurt.B, Kurt.V. (2013). Some Identities on the Generalized q-Bernoulli, q-Euler, and

q−Genocchi Polynomials. Abstract and Applied Analysis. 2013(293532), 6.

Luo Q.M. & Srivastava. M.H. (2006). Some relationships between the Apostol-Bernoulli and

Apostol-Euler-polynomials. Elsevier. 51(3–4), 631-642.

Luo, Q.M. (2010). Some results for the q-Bernoulli and q-Euler polynomials. Elsevier. 363(1),

7-18.

54

Naim.T. & KUS, S. (2015). Q-Bernoulli Matrices and Their Some Properties. Gazi University

Journal Of Science. 21(2).

Nalci.S. & Pashaev, O. K. (2012). q-Bernoulli Numbers and Zeros of q-Sine Function.

http://arxiv.org/abs/1202.2265v1.

Riordan, J. (1968). Combinatorial Identities. NewYork, London Sydney: Wiley.

Rudin, W. (1964). Principle Of Mathematical Induction. United State of America, U.S:

McGraw-Hill, Inc.

Srivastava, H.M. & pinter.A. (2004). Remarks on some relationships between the Bernoulli

and Euler polynomials. Applied Mathematics. 17(4), 375-380.

Srivastava.H.M. & Luo. (2006). Some relationships between the Apostol-Bernoulli and

Apostol-Euler-polynomials. Elsevier. 51(3–4), 631-642.

Srivastava.H.M. & Luo. (2011). q-Extensions of some relationships between the Bernoulli and

Euler polynomials. Taiwan J. Mathematics. 15(1)241-257.

Srivastava, H. M. & Vignat.C. (2012). Probabilistic proofs of some relationships between the

Bernoulli and Euler polynomials. European Journal and Applied Mathematics. 5(2).

Ernst. T. (2012). A comprehensive treatment of q-Calculus. United Kingdom: Springer Basel.

Kim.T. (2006). q-Generalized Euler numbers and polynomials. Springer. 13(3), 293–298.

Tsuneo, A., Tomoyoshi, I., Masanobu, K., and Don, Z. (2014). Bernoulli Numbers and Zeta

Function. Japan: Springer.

Victor. K. Cheung, & Pokman. (2002). Quantum Calculus. United state of America: Springer-

Verlag New York Berlin Heidelberg.

Zhizheng, Z. (1997). The Linear Algebra of the Generalized Pascal Matrix. Elsevier Science

Inc. 250(1), 51-60.

Zhizheng, Z., & JunWang. (2006). Bernoulli matrix and its algebraic properties. Elsevier B.V.

154(11), 1622–1632.

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56