Advances in Difference Equations

, 2013:115 | Cite as

On a class of generalized q-Bernoulli and q-Euler polynomials

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  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

Abstract

The main purpose of this paper is to introduce and investigate a new class of generalized q-Bernoulli and q-Euler polynomials. The q-analogues of well-known formulas are derived. A generalization of the Srivastava-Pintér addition theorem is obtained.

Keywords

Nonnegative Integer Correct Form Euler Number Basic Formula Addition Theorem 

1 Introduction

Throughout this paper, we always make use of the following notation: ℕ denotes the set of natural numbers, N 0 Open image in new window denotes the set of nonnegative integers, ℝ denotes the set of real numbers, ℂ denotes the set of complex numbers.

The q-numbers and q-factorial are defined by
[ a ] q = 1 q a 1 q ( q 1 ) ; [ 0 ] q ! = 1 ; [ n ] q ! = [ 1 ] q [ 2 ] q [ n ] q , n N , a C , Open image in new window
respectively. The q-polynomial coefficient is defined by
[ n k ] q = ( q ; q ) n ( q ; q ) n k ( q ; q ) k . Open image in new window
The q-analogue of the function ( x + y ) n Open image in new window is defined by
( x + y ) q n : = k = 0 n [ n k ] q q 1 2 k ( k 1 ) x n k y k , n N 0 . Open image in new window
The q-binomial formula is known as
( 1 a ) q n = j = 0 n 1 ( 1 q j a ) = k = 0 n [ n k ] q q 1 2 k ( k 1 ) ( 1 ) k a k . Open image in new window
In the standard approach to the q-calculus, two exponential functions are used:
e q ( z ) = n = 0 z n [ n ] q ! = k = 0 1 ( 1 ( 1 q ) q k z ) , 0 < | q | < 1 , | z | < 1 | 1 q | , E q ( z ) = n = 0 q 1 2 n ( n 1 ) z n [ n ] q ! = k = 0 ( 1 + ( 1 q ) q k z ) , 0 < | q | < 1 , z C . Open image in new window
From this form, we easily see that e q ( z ) E q ( z ) = 1 Open image in new window. Moreover,
D q e q ( z ) = e q ( z ) , D q E q ( z ) = E q ( q z ) , Open image in new window
where D q Open image in new window is defined by
D q f ( z ) : = f ( q z ) f ( z ) q z z , 0 < | q | < 1 , 0 z C . Open image in new window

The above q-standard notation can be found in [1].

Carlitz firstly extended the classical Bernoulli and Euler numbers and polynomials, introducing them as q-Bernoulli and q-Euler numbers and polynomials [2, 3, 4]. There are numerous recent investigations on this subject by, among many other authors, Cenki et al. [5, 6, 7], Choi et al. [8] and [9], Kim et al. [10, 11, 12, 13], Ozden and Simsek [14], Ryoo et al. [15], Simsek [16, 17] and [18], and Luo and Srivastava [19], Srivastava et al. [20], Mahmudov [21, 22].

Recently, Natalini and Bernardini [23], Bretti et al. [24], Kurt [25, 26], Tremblay et al. [27, 28] studied the properties of the following generalized Bernoulli and Euler polynomials:
( t m e t k = 0 m 1 t k k ! ) α e t x = n = 0 B n [ m 1 , α ] ( x ) t n n ! , ( t m e t + k = 0 m 1 t k k ! ) α e t x = n = 0 E n [ m 1 , α ] ( x ) t n n ! , α C , 1 α : = 1 . Open image in new window
(1)

Motivated by the generalizations in (1) of the classical Bernoulli and Euler polynomials, we introduce and investigate here the so-called generalized two-dimensional q-Bernoulli and q-Euler polynomials, which are defined as follows.

Definition 1 Let q , α C Open image in new window, m N Open image in new window, 0 < | q | < 1 Open image in new window. The generalized two-dimensional q-Bernoulli polynomials B n , q [ m 1 , α ] ( x , y ) Open image in new window are defined, in a suitable neighborhood of t = 0 Open image in new window, by means of the generating function
( t m e q ( t ) T m 1 , q ( t ) ) α e q ( t x ) E q ( t y ) = n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! , Open image in new window

where T m 1 , q ( t ) = k = 0 m 1 t k [ k ] q ! Open image in new window.

Definition 2 Let q , α C Open image in new window, 0 < | q | < 1 Open image in new window, m N Open image in new window. The generalized two-dimensional q-Euler polynomials E n , q [ m 1 , α ] ( x , y ) Open image in new window are defined, in a suitable neighborhood of t = 0 Open image in new window, by means of the generating functions
( 2 m e q ( t ) + T m 1 , q ( t ) ) α e q ( t x ) E q ( t y ) = n = 0 E n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! . Open image in new window
It is obvious that
lim q 1 B n , q [ m 1 , α ] ( x , y ) = B n [ m 1 , α ] ( x + y ) , B n , q [ m 1 , α ] = B n , q [ m 1 , α ] ( 0 , 0 ) , lim q 1 B n , q [ m 1 , α ] = B n [ m 1 , α ] , lim q 1 E n , q [ m 1 , α ] ( x , y ) = E n [ m 1 , α ] ( x + y ) , E n , q [ m 1 , α ] = E n , q [ m 1 , α ] ( 0 , 0 ) , lim q 1 E n , q [ m 1 , α ] = E n [ m 1 , α ] , lim q 1 B n , q [ m 1 , α ] ( x , 0 ) = B n [ m 1 , α ] ( x ) , lim q 1 B n , q [ m 1 , α ] ( 0 , y ) = B n [ m 1 , α ] ( y ) , lim q 1 E n , q [ m 1 , α ] ( x , 0 ) = E n [ m 1 , α ] ( x ) , lim q 1 E n , q [ m 1 , α ] ( 0 , y ) = E n [ m 1 , α ] ( y ) . Open image in new window

Here B n [ m 1 , α ] ( x ) Open image in new window and E n [ m 1 , α ] ( x ) Open image in new window denote the generalized Bernoulli and Euler polynomials defined in (1). Notice that B n [ m 1 , α ] ( x ) Open image in new window was introduced by Natalini [23], and E n [ m 1 , α ] ( x ) Open image in new window was introduced by Kurt [25].

In fact Definitions 1 and 2 define two different types B n , q [ m 1 , α ] ( x , 0 ) Open image in new window and B n , q [ m 1 , α ] ( 0 , y ) Open image in new window of the generalized q-Bernoulli polynomials and two different types E n , q [ m 1 , α ] ( x , 0 ) Open image in new window and E n , q [ m 1 , α ] ( 0 , y ) Open image in new window of the generalized q-Euler polynomials. Both polynomials B n , q [ m 1 , α ] ( x , 0 ) Open image in new window and B n , q [ m 1 , α ] ( 0 , y ) Open image in new window ( E n , q [ m 1 , α ] ( x , 0 ) Open image in new window and E n , q [ m 1 , α ] ( 0 , y ) Open image in new window) coincide with the classical higher-order generalized Bernoulli polynomials (Euler polynomials) in the limiting case q 1 Open image in new window.

2 Preliminaries and lemmas

In this section we provide some basic formulas for the generalized q-Bernoulli and q-Euler polynomials to obtain the main results of this paper in the next section. The following result is a q-analogue of the addition theorem for the classical Bernoulli and Euler polynomials.

Lemma 3 For all x , y C Open image in new window we have
B n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( x + y ) q n k , E n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( x + y ) q n k , Open image in new window
(2)
B n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 B k , q [ m 1 , α ] ( x , 0 ) y n k = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( 0 , y ) x n k , Open image in new window
(3)
E n , q [ m 1 , α ] ( x , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 E k , q [ m 1 , α ] ( x , 0 ) y n k = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( 0 , y ) x n k . Open image in new window
(4)
In particular, setting x = 0 Open image in new window and y = 0 Open image in new window in (3) and (4), we get the following formulae for the generalized q-Bernoulli and q-Euler polynomials, respectively,
B n , q [ m 1 , α ] ( x , 0 ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] x n k , B n , q [ m 1 , α ] ( 0 , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 B k , q [ m 1 , α ] y n k , E n , q [ m 1 , α ] ( x , 0 ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] x n k , E n , q [ m 1 , α ] ( 0 , y ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 E k , q [ m 1 , α ] y n k . Open image in new window
Setting y = 1 Open image in new window and x = 1 Open image in new window in (3) and (4), we get, respectively,
B n , q [ m 1 , α ] ( x , 1 ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 B k , q [ m 1 , α ] ( x , 0 ) , B n , q [ m 1 , α ] ( 1 , y ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( 0 , y ) , Open image in new window
(5)
E n , q [ m 1 , α ] ( x , 1 ) = k = 0 n [ n k ] q q ( n k ) ( n k 1 ) / 2 E k , q ( α ) ( x , 0 ) , E n , q [ m 1 , α ] ( 1 , y ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( 0 , y ) . Open image in new window
(6)
Clearly, (5) and (6) are the generalization of q-analogues of
B n ( x + 1 ) = k = 0 n ( n k ) B k ( x ) , E n ( x + 1 ) = k = 0 n ( n k ) E k ( x ) , Open image in new window

respectively.

Lemma 4 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:
B n , q [ m 1 , α + β ] ( x , y ) = k = 0 n [ n k ] q B k , q [ m 1 , α ] ( x , 0 ) B k , q [ m 1 , β ] ( 0 , y ) , E n , q [ m 1 , α + β ] ( x , y ) = k = 0 n [ n k ] q E k , q [ m 1 , α ] ( x , 0 ) E k , q [ m 1 , β ] ( 0 , y ) . Open image in new window
Lemma 5 We have
D q , x B n , q [ m 1 , α ] ( x , y ) = [ n ] q B n 1 , q [ m 1 , α ] ( x , y ) , D q , y B n , q [ m 1 , α ] ( x , y ) = [ n ] q B n 1 , q [ m 1 , α ] ( x , q y ) , D q , x E n , q [ m 1 , α ] ( x , y ) = [ n ] q E n 1 , q [ m 1 , α ] ( x , y ) , D q , y E n , q [ m 1 , α ] ( x , y ) = [ n ] q E n 1 , q [ m 1 , α ] ( x , q y ) . Open image in new window
Lemma 6 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:
B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) = [ n ] q ! [ n m ] q ! B n m , q [ m 1 , α 1 ] ( 0 , y ) , n m , E n , q [ m 1 , α ] ( 1 , y ) + k = 0 min ( n , m 1 ) [ n k ] q E n , q [ m 1 , α ] ( 0 , y ) = 2 m E n , q [ m 1 , α 1 ] ( 0 , y ) , B n , q [ m 1 , α ] ( x , 0 ) k = 0 min ( n , m 1 ) [ n k ] q B n , q [ m 1 , α ] ( x , 1 ) = [ n ] q ! [ n m ] q ! B n m , q [ m 1 , α 1 ] ( x , 1 ) , n m , E n , q [ m 1 , α ] ( x , 0 ) + k = 0 min ( n , m 1 ) [ n k ] q E n , q [ m 1 , α ] ( x , 1 ) = 2 m E n , q [ m 1 , α 1 ] ( x , 1 ) . Open image in new window
(7)
Proof We prove only (7). The proof is based on the following equality:
n = 0 ( B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) ) t n [ n ] q ! = ( t m e q ( t ) T m 1 , q ( t ) ) α e q ( t ) E q ( t y ) T m 1 , q ( t ) ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) = ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) ( e q ( t ) T m 1 , q ( t ) ) = t m ( t m e q ( t ) T m 1 , q ( t ) ) α 1 E q ( t y ) = n = 0 [ n + m ] q ! [ n ] q ! B n , q [ m 1 , α 1 ] ( 0 , y ) t n + m [ n + m ] q ! . Open image in new window
Here we used the following relation:
T m 1 , q ( t ) ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) = n = 0 m 1 t n [ n ] q ! n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! = n = 0 B n , q [ m 1 , α ] ( 0 , y ) ( t n [ n ] q ! + t n + 1 [ n ] q ! + t n + 2 [ n ] q ! [ 2 ] q ! + + t n + m 1 [ n ] q ! [ m 1 ] q ! ) = n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! + n = 0 [ n ] q B n 1 , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! + n = 0 [ n ] q [ n 1 ] q [ 2 ] q ! B n 2 , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! + + n = 0 [ n ] q [ n m + 2 ] q [ m 1 ] q ! B n m + 1 , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! = n = 0 k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! . Open image in new window

 □

Corollary 7 Taking q 1 Open image in new window, we have
B n [ m 1 , α ] ( y + 1 ) k = 0 min ( n , m 1 ) [ n k ] q B n k [ m 1 , α ] ( y ) = [ n ] q ! [ n m ] q ! B n m [ m 1 , α 1 ] ( y ) , n m , E n [ m 1 , α ] ( y + 1 ) + k = 0 min ( n , m 1 ) [ n k ] q E n [ m 1 , α ] ( y ) = 2 m E n [ m 1 , α 1 ] ( y ) . Open image in new window
Lemma 8 The generalized q-Bernoulli polynomials satisfy the following relations:
B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) = [ n ] q k = 0 n 1 [ n 1 k ] q B k , q [ m 1 , α ] ( 0 , y ) B n 1 k , q [ 0 , 1 ] . Open image in new window
(8)

Proof

Indeed,
n = 0 ( B n , q [ m 1 , α ] ( 1 , y ) k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) ) t n [ n ] q ! = ( t m e q ( t ) T m 1 , q ( t ) ) α e q ( t ) E q ( t y ) T m 1 , q ( t ) ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) = ( t m e q ( t ) T m 1 , q ( t ) ) α E q ( t y ) e q ( t ) T m 1 , q ( t ) t t = n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! n = 0 B n , q [ 0 , 1 ] t n + 1 [ n ] q ! = n = 1 [ n ] q k = 0 n 1 [ n 1 k ] q B k , q [ m 1 , α ] ( 0 , y ) B n 1 k , q [ 0 , 1 ] t n [ n ] q ! . Open image in new window

 □

Remark 9 Notice taking limit in (8) as q 1 Open image in new window, we get
B n [ m 1 , α ] ( y + 1 ) k = 0 min ( n , m 1 ) ( n k ) B n k [ m 1 , α ] ( y ) = n k = 0 n 1 ( n 1 k ) B k [ m 1 , α ] ( y ) B n 1 k [ 0 , 1 ] . Open image in new window

It is a correct form of formula (2.7) from [27] for λ = 1 Open image in new window.

Lemma 10 We have
x n = k = 0 n [ n k ] q [ k ] q ! [ k + m ] q ! B n k , q [ m 1 , 1 ] ( x , 0 ) , y n = 1 q n ( n 1 ) 2 k = 0 n [ n k ] q [ k ] q ! [ k + m ] q ! B n k , q [ m 1 , 1 ] ( 0 , y ) , x n = 1 2 m ( k = 0 n [ n k ] q E k , q [ m 1 , 1 ] ( x , 0 ) + k = 0 min ( n , m 1 ) [ n k ] q E k , q [ m 1 , 1 ] ( x , 0 ) ) , y n = 1 2 m q n ( n 1 ) 2 ( k = 0 n [ n k ] q E k , q [ m 1 , 1 ] ( 0 , y ) + k = 0 min ( n , m 1 ) [ n k ] q E n , q [ m 1 , 1 ] ( 0 , y ) ) . Open image in new window
From Lemma 10 we obtain the list of generalized q-Bernoulli polynomials as follows
B 0 , q [ m 1 , 1 ] ( x , 0 ) = [ m ] q ! , B 0 , q [ m 1 , 1 ] ( 0 , y ) = [ m ] q ! , B 1 , q [ m 1 , 1 ] ( x , 0 ) = [ m ] q ! ( x 1 [ m + 1 ] q ) , B 1 , q [ m 1 , 1 ] ( 0 , y ) = [ m ] q ! ( y 1 [ m + 1 ] q ) , B 2 , q [ m 1 , 1 ] ( x , 0 ) = x 2 [ 2 ] q [ m ] q ! [ m + 1 ] q x + [ 2 ] q q m + 1 [ m ] q ! [ m + 1 ] q 2 [ m + 2 ] q , B 2 , q [ m 1 , 1 ] ( 0 , y ) = q y 2 [ 2 ] q [ m ] q ! [ m + 1 ] q y + [ 2 ] q q m + 1 [ m ] q ! [ m + 1 ] q 2 [ m + 2 ] q . Open image in new window

3 Explicit relationship between the q-Bernoulli and q-Euler polynomials

In this section, we give some generalizations of the Srivastava-Pintér addition theorem. We also obtain new formulae and their some special cases below.

We present natural q-extensions of the main results of the papers [29, 30].

Theorem 11 The relationships
B n , q [ m 1 , α ] ( x , y ) = 1 2 k = 0 n [ n k ] q [ 1 l n k B k , q [ m 1 , α ] ( x , 0 ) + j = 0 k [ k j ] q 1 l k j B j , q [ m 1 , α ] ( x , 0 ) ] E n k , q ( 0 , l y ) , Open image in new window
(9)
B n , q [ m 1 , α ] ( x , y ) = 1 2 k = 0 n [ n k ] q 1 l n k [ B k , q [ m 1 , α ] ( 0 , y ) + B k , q [ m 1 , α ] ( 1 l , y ) ] E n k , q ( l x , 0 ) Open image in new window
(10)

hold true between the generalized q-Bernoulli polynomials and q-Euler polynomials.

Proof First we prove (9). Using the identity
( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α e q ( t x ) E q ( t y ) = 2 e q ( t l ) + 1 E q ( t l l y ) e q ( t l ) + 1 2 ( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α e q ( t x ) , Open image in new window
we have
n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 E n , q ( 0 , l y ) t n l n [ n ] q ! k = 0 t k l k [ k ] q ! j = 0 B j , q [ m 1 , α ] ( x , 0 ) t j [ j ] q ! + 1 2 k = 0 E k , q ( 0 , l y ) t k l k [ k ] q ! n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! = : I 1 + I 2 . Open image in new window
It is clear that
I 2 = 1 2 n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! k = 0 E k , q ( 0 , l y ) t k l k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n B k , q [ m 1 , α ] ( x , 0 ) E n k , q ( 0 , l y ) t n [ n ] q ! . Open image in new window
On the other hand,
I 1 = 1 2 n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! k = 0 E k , q ( 0 , l y ) t k l k [ k ] q ! j = 0 t j l j [ j ] q ! = 1 2 n = 0 B n , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! k = 0 j = 0 k [ k j ] q E j , q ( 0 , l y ) t k l k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q B k , q [ m 1 , α ] ( x , 0 ) j = 0 n k [ n k j ] q 1 l n k E j , q ( 0 , l y ) t n [ n ] q ! = 1 2 n = 0 j = 0 n [ n j ] q E j , q ( 0 , l y ) k = 0 n j [ n j k ] q 1 l n k B k , q [ m 1 , α ] ( x , 0 ) t n [ n ] q ! . Open image in new window
Therefore
n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 k = 0 n [ n k ] q [ 1 l n k B k , q [ m 1 , α ] ( x , 0 ) + j = 0 k [ k j ] q 1 l k j B j , q [ m 1 , α ] ( x , 0 ) ] × E n k , q ( 0 , l y ) t n [ n ] q ! . Open image in new window
Next we prove (10). Using the identity
( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α e q ( t x ) E q ( t y ) = 2 e q ( t l ) + 1 e q ( t l l x ) e q ( t l ) + 1 2 ( t m e q ( t ) i = 0 m 1 t i [ i ] q ! ) α E q ( t y ) , Open image in new window
we have
n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 E n , q ( l x , 0 ) t n l n [ n ] q ! n = 0 B n , q [ m 1 , α ] ( 1 l , y ) t n [ n ] q ! + 1 2 k = 0 E k , q ( l x , 0 ) t k l k [ k ] q ! n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! = : I 1 + I 2 . Open image in new window
It is clear that
I 2 = 1 2 n = 0 B n , q [ m 1 , α ] ( 0 , y ) t n [ n ] q ! k = 0 E k , q ( l x , 0 ) t k l k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n B k , q [ m 1 , α ] ( 0 , y ) E n k , q ( l x , 0 ) t n [ n ] q ! . Open image in new window
On the other hand,
I 1 = 1 2 n = 0 B n , q [ m 1 , α ] ( 1 l , y ) t n [ n ] q ! k = 0 E k , q ( l x , 0 ) t k m k [ k ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n B k , q [ m 1 , α ] ( 1 l , y ) E n k , q ( l x , 0 ) t n [ n ] q ! . Open image in new window
Therefore
n = 0 B n , q [ m 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 n = 0 k = 0 n [ n k ] q l k n [ B k , q [ m 1 , α ] ( 0 , y ) + B k , q [ m 1 , α ] ( 1 l , y ) ] E n k , q ( l x , 0 ) t n [ n ] q ! . Open image in new window

 □

Next we discuss some special cases of Theorem 11.

Theorem 12 The relationship
B n , q [ m 1 , α ] ( x , y ) = 1 2 k = 0 n [ n k ] q [ B k , q [ m 1 , α ] ( 0 , y ) + k = 0 min ( n , m 1 ) [ n k ] q B n k , q [ m 1 , α ] ( 0 , y ) + [ k ] q j = 0 k 1 [ k 1 j ] q B j , q [ m 1 , α ] ( 0 , y ) B k 1 j , q [ 0 , 1 ] ] E n k , q ( x , 0 ) Open image in new window

holds true between the generalized q-Bernoulli polynomials and the q-Euler polynomials.

Remark 13 Taking q 1 Open image in new window in Theorem 12, we obtain the Srivastava-Pintér addition theorem for the generalized Bernoulli and Euler polynomials.
B n [ m 1 , α ] ( x + y ) = 1 2 k = 0 n ( n k ) [ B k [ m 1 , α ] ( y ) + k = 0 min ( n , m 1 ) ( n k ) B n k [ m 1 , α ] ( y ) + k j = 0 k 1 ( k 1 j ) B j [ m 1 , α ] ( y ) B k 1 j [ 0 , 1 ] ] E n k ( x ) . Open image in new window
(11)

Notice that the Srivastava-Pintér addition theorem for the generalized Apostol-Bernoulli polynomials and the Apostol-Euler polynomials was given in [27]. The formula (11) is a correct version of Theorem 3 [27] for λ = 1 Open image in new window.

Notes

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the reviewers for their valuable comments and helpful suggestions that improved the note’s quality.

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Copyright information

© Mahmudov and Keleshteri; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Eastern Mediterranean UniversityGazimagusaTurkey

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