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Advances in Difference Equations

, 2013:115 | Cite as

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

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Research
Part of the following topical collections:
  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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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