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

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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.

1 Introduction

Throughout this paper, we always make use of the following notation: denotes the set of natural numbers, $N 0$ denotes the set of nonnegative integers, denotes the set of real numbers, denotes the set of complex numbers.

The q-numbers and q-factorial are defined by

$[ a ] q = 1 − q a 1 − q (q≠1); [ 0 ] q !=1; [ n ] q != [ 1 ] q [ 2 ] q ⋯ [ n ] q ,n∈N,a∈C,$

respectively. The q-polynomial coefficient is defined by

$[ n k ] q = ( q ; q ) n ( q ; q ) n − k ( q ; q ) k .$

The q-analogue of the function $( x + y ) n$ is defined by

$( x + y ) q n := ∑ k = 0 n [ n k ] q q 1 2 k ( k − 1 ) x n − k y k ,n∈ N 0 .$

The q-binomial formula is known as

$( 1 − a ) q n = ∏ j = 0 n − 1 ( 1 − q j a ) = ∑ k = 0 n [ n k ] q q 1 2 k ( k − 1 ) ( − 1 ) k a k .$

In the standard approach to the q-calculus, two exponential functions are used:

$e q ( z ) = ∑ n = 0 ∞ z n [ n ] q ! = ∏ k = 0 ∞ 1 ( 1 − ( 1 − q ) q k z ) , 0 < | q | < 1 , | z | < 1 | 1 − q | , E q ( z ) = ∑ n = 0 ∞ q 1 2 n ( n − 1 ) z n [ n ] q ! = ∏ k = 0 ∞ ( 1 + ( 1 − q ) q k z ) , 0 < | q | < 1 , z ∈ C .$

From this form, we easily see that $e q (z) E q (−z)=1$. Moreover,

$D q e q (z)= e q (z), D q E q (z)= E q (qz),$

where $D q$ is defined by

$D q f(z):= f ( q z ) − f ( z ) q z − z ,0<|q|<1,0≠z∈C.$

The above q-standard notation can be found in .

Carlitz firstly extended the classical Bernoulli and Euler numbers and polynomials, introducing them as q-Bernoulli and q-Euler numbers and polynomials . There are numerous recent investigations on this subject by, among many other authors, Cenki et al. , Choi et al.  and , Kim et al. , Ozden and Simsek , Ryoo et al. , Simsek [16, 17] and , and Luo and Srivastava , Srivastava et al. , Mahmudov [21, 22].

Recently, Natalini and Bernardini , Bretti et al. , Kurt [25, 26], Tremblay et al. [27, 28] studied the properties of the following generalized Bernoulli and Euler polynomials:

$( t m e t − ∑ k = 0 m − 1 t k k ! ) α e t x = ∑ n = 0 ∞ B n [ m − 1 , α ] ( x ) t n n ! , ( t m e t + ∑ k = 0 m − 1 t k k ! ) α e t x = ∑ n = 0 ∞ E n [ m − 1 , α ] ( x ) t n n ! , α ∈ C , 1 α : = 1 .$
(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 $q,α∈C$, $m∈N$, $0<|q|<1$. The generalized two-dimensional q-Bernoulli polynomials $B n , q [ m − 1 , α ] (x,y)$ are defined, in a suitable neighborhood of $t=0$, by means of the generating function

$( t m e q ( t ) − T m − 1 , q ( t ) ) α e q (tx) E q (ty)= ∑ n = 0 ∞ B n , q [ m − 1 , α ] (x,y) t n [ n ] q ! ,$

where $T m − 1 , q (t)= ∑ k = 0 m − 1 t k [ k ] q !$.

Definition 2 Let $q,α∈C$, $0<|q|<1$, $m∈N$. The generalized two-dimensional q-Euler polynomials $E n , q [ m − 1 , α ] (x,y)$ are defined, in a suitable neighborhood of $t=0$, by means of the generating functions

$( 2 m e q ( t ) + T m − 1 , q ( t ) ) α e q (tx) E q (ty)= ∑ n = 0 ∞ E n , q [ m − 1 , α ] (x,y) t n [ n ] q ! .$

It is obvious that

$lim q → 1 − B n , q [ m − 1 , α ] ( x , y ) = B n [ m − 1 , α ] ( x + y ) , B n , q [ m − 1 , α ] = B n , q [ m − 1 , α ] ( 0 , 0 ) , lim q → 1 − B n , q [ m − 1 , α ] = B n [ m − 1 , α ] , lim q → 1 − E n , q [ m − 1 , α ] ( x , y ) = E n [ m − 1 , α ] ( x + y ) , E n , q [ m − 1 , α ] = E n , q [ m − 1 , α ] ( 0 , 0 ) , lim q → 1 − E n , q [ m − 1 , α ] = E n [ m − 1 , α ] , lim q → 1 − B n , q [ m − 1 , α ] ( x , 0 ) = B n [ m − 1 , α ] ( x ) , lim q → 1 − B n , q [ m − 1 , α ] ( 0 , y ) = B n [ m − 1 , α ] ( y ) , lim q → 1 − E n , q [ m − 1 , α ] ( x , 0 ) = E n [ m − 1 , α ] ( x ) , lim q → 1 − E n , q [ m − 1 , α ] ( 0 , y ) = E n [ m − 1 , α ] ( y ) .$

Here $B n [ m − 1 , α ] (x)$ and $E n [ m − 1 , α ] (x)$ denote the generalized Bernoulli and Euler polynomials defined in (1). Notice that $B n [ m − 1 , α ] (x)$ was introduced by Natalini , and $E n [ m − 1 , α ] (x)$ was introduced by Kurt .

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

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 $x,y∈C$ we have

$B n , q [ m − 1 , α ] ( x , y ) = ∑ k = 0 n [ n k ] q B k , q [ m − 1 , α ] ( x + y ) q n − k , E n , q [ m − 1 , α ] ( x , y ) = ∑ k = 0 n [ n k ] q E k , q [ m − 1 , α ] ( x + y ) q n − k ,$
(2)
$B n , q [ m − 1 , α ] ( x , y ) = ∑ k = 0 n [ n k ] q q ( n − k ) ( n − k − 1 ) / 2 B k , q [ m − 1 , α ] ( x , 0 ) y n − k = ∑ k = 0 n [ n k ] q B k , q [ m − 1 , α ] ( 0 , y ) x n − k ,$
(3)
$E n , q [ m − 1 , α ] ( x , y ) = ∑ k = 0 n [ n k ] q q ( n − k ) ( n − k − 1 ) / 2 E k , q [ m − 1 , α ] ( x , 0 ) y n − k = ∑ k = 0 n [ n k ] q E k , q [ m − 1 , α ] ( 0 , y ) x n − k .$
(4)

In particular, setting $x=0$ and $y=0$ in (3) and (4), we get the following formulae for the generalized q-Bernoulli and q-Euler polynomials, respectively,

$B n , q [ m − 1 , α ] ( x , 0 ) = ∑ k = 0 n [ n k ] q B k , q [ m − 1 , α ] x n − k , B n , q [ m − 1 , α ] ( 0 , y ) = ∑ k = 0 n [ n k ] q q ( n − k ) ( n − k − 1 ) / 2 B k , q [ m − 1 , α ] y n − k , E n , q [ m − 1 , α ] ( x , 0 ) = ∑ k = 0 n [ n k ] q E k , q [ m − 1 , α ] x n − k , E n , q [ m − 1 , α ] ( 0 , y ) = ∑ k = 0 n [ n k ] q q ( n − k ) ( n − k − 1 ) / 2 E k , q [ m − 1 , α ] y n − k .$

Setting $y=1$ and $x=1$ in (3) and (4), we get, respectively,

$B n , q [ m − 1 , α ] ( x , 1 ) = ∑ k = 0 n [ n k ] q q ( n − k ) ( n − k − 1 ) / 2 B k , q [ m − 1 , α ] ( x , 0 ) , B n , q [ m − 1 , α ] ( 1 , y ) = ∑ k = 0 n [ n k ] q B k , q [ m − 1 , α ] ( 0 , y ) ,$
(5)
$E n , q [ m − 1 , α ] ( x , 1 ) = ∑ k = 0 n [ n k ] q q ( n − k ) ( n − k − 1 ) / 2 E k , q ( α ) ( x , 0 ) , E n , q [ m − 1 , α ] ( 1 , y ) = ∑ k = 0 n [ n k ] q E k , q [ m − 1 , α ] ( 0 , y ) .$
(6)

Clearly, (5) and (6) are the generalization of q-analogues of

$B n (x+1)= ∑ k = 0 n ( n k ) B k (x), E n (x+1)= ∑ k = 0 n ( n k ) E k (x),$

respectively.

Lemma 4 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:

$B n , q [ m − 1 , α + β ] ( x , y ) = ∑ k = 0 n [ n k ] q B k , q [ m − 1 , α ] ( x , 0 ) B k , q [ m − 1 , β ] ( 0 , y ) , E n , q [ m − 1 , α + β ] ( x , y ) = ∑ k = 0 n [ n k ] q E k , q [ m − 1 , α ] ( x , 0 ) E k , q [ m − 1 , β ] ( 0 , y ) .$

Lemma 5 We have

$D q , x B n , q [ m − 1 , α ] ( x , y ) = [ n ] q B n − 1 , q [ m − 1 , α ] ( x , y ) , D q , y B n , q [ m − 1 , α ] ( x , y ) = [ n ] q B n − 1 , q [ m − 1 , α ] ( x , q y ) , D q , x E n , q [ m − 1 , α ] ( x , y ) = [ n ] q E n − 1 , q [ m − 1 , α ] ( x , y ) , D q , y E n , q [ m − 1 , α ] ( x , y ) = [ n ] q E n − 1 , q [ m − 1 , α ] ( x , q y ) .$

Lemma 6 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:

$B n , q [ m − 1 , α ] ( 1 , y ) − ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k , q [ m − 1 , α ] ( 0 , y ) = [ n ] q ! [ n − m ] q ! B n − m , q [ m − 1 , α − 1 ] ( 0 , y ) , n ≥ m , E n , q [ m − 1 , α ] ( 1 , y ) + ∑ k = 0 min ( n , m − 1 ) [ n k ] q E n , q [ m − 1 , α ] ( 0 , y ) = 2 m E n , q [ m − 1 , α − 1 ] ( 0 , y ) , B n , q [ m − 1 , α ] ( x , 0 ) − ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n , q [ m − 1 , α ] ( x , − 1 ) = [ n ] q ! [ n − m ] q ! B n − m , q [ m − 1 , α − 1 ] ( x , − 1 ) , n ≥ m , E n , q [ m − 1 , α ] ( x , 0 ) + ∑ k = 0 min ( n , m − 1 ) [ n k ] q E n , q [ m − 1 , α ] ( x , − 1 ) = 2 m E n , q [ m − 1 , α − 1 ] ( x , − 1 ) .$
(7)

Proof We prove only (7). The proof is based on the following equality:

$∑ n = 0 ∞ ( B n , q [ m − 1 , α ] ( 1 , y ) − ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k , q [ m − 1 , α ] ( 0 , y ) ) t n [ n ] q ! = ( t m e q ( t ) − T m − 1 , q ( t ) ) α e q ( t ) E q ( t y ) − T m − 1 , q ( t ) ( t m e q ( t ) − T m − 1 , q ( t ) ) α E q ( t y ) = ( t m e q ( t ) − T m − 1 , q ( t ) ) α E q ( t y ) ( e q ( t ) − T m − 1 , q ( t ) ) = t m ( t m e q ( t ) − T m − 1 , q ( t ) ) α − 1 E q ( t y ) = ∑ n = 0 ∞ [ n + m ] q ! [ n ] q ! B n , q [ m − 1 , α − 1 ] ( 0 , y ) t n + m [ n + m ] q ! .$

Here we used the following relation:

$T m − 1 , q ( t ) ( t m e q ( t ) − T m − 1 , q ( t ) ) α E q ( t y ) = ∑ n = 0 m − 1 t n [ n ] q ! ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! = ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 0 , y ) ( t n [ n ] q ! + t n + 1 [ n ] q ! + t n + 2 [ n ] q ! [ 2 ] q ! + ⋯ + t n + m − 1 [ n ] q ! [ m − 1 ] q ! ) = ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! + ∑ n = 0 ∞ [ n ] q B n − 1 , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! + ∑ n = 0 ∞ [ n ] q [ n − 1 ] q [ 2 ] q ! B n − 2 , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! + ⋯ + ∑ n = 0 ∞ [ n ] q ⋯ [ n − m + 2 ] q [ m − 1 ] q ! B n − m + 1 , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! = ∑ n = 0 ∞ ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! .$

□

Corollary 7 Taking $q→ 1 −$, we have

$B n [ m − 1 , α ] ( y + 1 ) − ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k [ m − 1 , α ] ( y ) = [ n ] q ! [ n − m ] q ! B n − m [ m − 1 , α − 1 ] ( y ) , n ≥ m , E n [ m − 1 , α ] ( y + 1 ) + ∑ k = 0 min ( n , m − 1 ) [ n k ] q E n [ m − 1 , α ] ( y ) = 2 m E n [ m − 1 , α − 1 ] ( y ) .$

Lemma 8 The generalized q-Bernoulli polynomials satisfy the following relations:

$B n , q [ m − 1 , α ] ( 1 , y ) − ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k , q [ m − 1 , α ] ( 0 , y ) = [ n ] q ∑ k = 0 n − 1 [ n − 1 k ] q B k , q [ m − 1 , α ] ( 0 , y ) B n − 1 − k , q [ 0 , − 1 ] .$
(8)

Proof

Indeed,

$∑ n = 0 ∞ ( B n , q [ m − 1 , α ] ( 1 , y ) − ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k , q [ m − 1 , α ] ( 0 , y ) ) t n [ n ] q ! = ( t m e q ( t ) − T m − 1 , q ( t ) ) α e q ( t ) E q ( t y ) − T m − 1 , q ( t ) ( t m e q ( t ) − T m − 1 , q ( t ) ) α E q ( t y ) = ( t m e q ( t ) − T m − 1 , q ( t ) ) α E q ( t y ) e q ( t ) − T m − 1 , q ( t ) t t = ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! ∑ n = 0 ∞ B n , q [ 0 , − 1 ] t n + 1 [ n ] q ! = ∑ n = 1 ∞ [ n ] q ∑ k = 0 n − 1 [ n − 1 k ] q B k , q [ m − 1 , α ] ( 0 , y ) B n − 1 − k , q [ 0 , − 1 ] t n [ n ] q ! .$

□

Remark 9 Notice taking limit in (8) as $q→ 1 −$, we get

$B n [ m − 1 , α ] (y+1)− ∑ k = 0 min ( n , m − 1 ) ( n k ) B n − k [ m − 1 , α ] (y)=n ∑ k = 0 n − 1 ( n − 1 k ) B k [ m − 1 , α ] (y) B n − 1 − k [ 0 , − 1 ] .$

It is a correct form of formula (2.7) from  for $λ=1$.

Lemma 10 We have

$x n = ∑ k = 0 n [ n k ] q [ k ] q ! [ k + m ] q ! B n − k , q [ m − 1 , 1 ] ( x , 0 ) , y n = 1 q n ( n − 1 ) 2 ∑ k = 0 n [ n k ] q [ k ] q ! [ k + m ] q ! B n − k , q [ m − 1 , 1 ] ( 0 , y ) , x n = 1 2 m ( ∑ k = 0 n [ n k ] q E k , q [ m − 1 , 1 ] ( x , 0 ) + ∑ k = 0 min ( n , m − 1 ) [ n k ] q E k , q [ m − 1 , 1 ] ( x , 0 ) ) , y n = 1 2 m q n ( n − 1 ) 2 ( ∑ k = 0 n [ n k ] q E k , q [ m − 1 , 1 ] ( 0 , y ) + ∑ k = 0 min ( n , m − 1 ) [ n k ] q E n , q [ m − 1 , 1 ] ( 0 , y ) ) .$

From Lemma 10 we obtain the list of generalized q-Bernoulli polynomials as follows

$B 0 , q [ m − 1 , 1 ] ( x , 0 ) = [ m ] q ! , B 0 , q [ m − 1 , 1 ] ( 0 , y ) = [ m ] q ! , B 1 , q [ m − 1 , 1 ] ( x , 0 ) = [ m ] q ! ( x − 1 [ m + 1 ] q ) , B 1 , q [ m − 1 , 1 ] ( 0 , y ) = [ m ] q ! ( y − 1 [ m + 1 ] q ) , B 2 , q [ m − 1 , 1 ] ( x , 0 ) = x 2 − [ 2 ] q [ m ] q ! [ m + 1 ] q x + [ 2 ] q q m + 1 [ m ] q ! [ m + 1 ] q 2 [ m + 2 ] q , B 2 , q [ m − 1 , 1 ] ( 0 , y ) = q y 2 − [ 2 ] q [ m ] q ! [ m + 1 ] q y + [ 2 ] q q m + 1 [ m ] q ! [ m + 1 ] q 2 [ m + 2 ] q .$

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

$B n , q [ m − 1 , α ] ( x , y ) = 1 2 ∑ k = 0 n [ n k ] q [ 1 l n − k B k , q [ m − 1 , α ] ( x , 0 ) + ∑ j = 0 k [ k j ] q 1 l k − j B j , q [ m − 1 , α ] ( x , 0 ) ] E n − k , q ( 0 , l y ) ,$
(9)
$B n , q [ m − 1 , α ] (x,y)= 1 2 ∑ k = 0 n [ n k ] q 1 l n − k [ B k , q [ m − 1 , α ] ( 0 , y ) + B k , q [ m − 1 , α ] ( 1 l , y ) ] E n − k , q (lx,0)$
(10)

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

Proof First we prove (9). Using the identity

$( t m e q ( t ) − ∑ i = 0 m − 1 t i [ i ] q ! ) α e q ( t x ) E q ( t y ) = 2 e q ( t l ) + 1 ⋅ E q ( t l l y ) ⋅ e q ( t l ) + 1 2 ⋅ ( t m e q ( t ) − ∑ i = 0 m − 1 t i [ i ] q ! ) α e q ( t x ) ,$

we have

$∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 ∑ n = 0 ∞ E n , q ( 0 , l y ) t n l n [ n ] q ! ∑ k = 0 ∞ t k l k [ k ] q ! ∑ j = 0 ∞ B j , q [ m − 1 , α ] ( x , 0 ) t j [ j ] q ! + 1 2 ∑ k = 0 ∞ E k , q ( 0 , l y ) t k l k [ k ] q ! ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , 0 ) t n [ n ] q ! = : I 1 + I 2 .$

It is clear that

$I 2 = 1 2 ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , 0 ) t n [ n ] q ! ∑ k = 0 ∞ E k , q ( 0 , l y ) t k l k [ k ] q ! = 1 2 ∑ n = 0 ∞ ∑ k = 0 n [ n k ] q l k − n B k , q [ m − 1 , α ] ( x , 0 ) E n − k , q ( 0 , l y ) t n [ n ] q ! .$

On the other hand,

$I 1 = 1 2 ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , 0 ) t n [ n ] q ! ∑ k = 0 ∞ E k , q ( 0 , l y ) t k l k [ k ] q ! ∑ j = 0 ∞ t j l j [ j ] q ! = 1 2 ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , 0 ) t n [ n ] q ! ∑ k = 0 ∞ ∑ j = 0 k [ k j ] q E j , q ( 0 , l y ) t k l k [ k ] q ! = 1 2 ∑ n = 0 ∞ ∑ k = 0 n [ n k ] q B k , q [ m − 1 , α ] ( x , 0 ) ∑ j = 0 n − k [ n − k j ] q 1 l n − k E j , q ( 0 , l y ) t n [ n ] q ! = 1 2 ∑ n = 0 ∞ ∑ j = 0 n [ n j ] q E j , q ( 0 , l y ) ∑ k = 0 n − j [ n − j k ] q 1 l n − k B k , q [ m − 1 , α ] ( x , 0 ) t n [ n ] q ! .$

Therefore

$∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 ∑ n = 0 ∞ ∑ k = 0 n [ n k ] q [ 1 l n − k B k , q [ m − 1 , α ] ( x , 0 ) + ∑ j = 0 k [ k j ] q 1 l k − j B j , q [ m − 1 , α ] ( x , 0 ) ] × E n − k , q ( 0 , l y ) t n [ n ] q ! .$

Next we prove (10). Using the identity

$( t m e q ( t ) − ∑ i = 0 m − 1 t i [ i ] q ! ) α e q ( t x ) E q ( t y ) = 2 e q ( t l ) + 1 ⋅ e q ( t l l x ) ⋅ e q ( t l ) + 1 2 ⋅ ( t m e q ( t ) − ∑ i = 0 m − 1 t i [ i ] q ! ) α E q ( t y ) ,$

we have

$∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 ∑ n = 0 ∞ E n , q ( l x , 0 ) t n l n [ n ] q ! ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 1 l , y ) t n [ n ] q ! + 1 2 ∑ k = 0 ∞ E k , q ( l x , 0 ) t k l k [ k ] q ! ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! = : I 1 + I 2 .$

It is clear that

$I 2 = 1 2 ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 0 , y ) t n [ n ] q ! ∑ k = 0 ∞ E k , q ( l x , 0 ) t k l k [ k ] q ! = 1 2 ∑ n = 0 ∞ ∑ k = 0 n [ n k ] q l k − n B k , q [ m − 1 , α ] ( 0 , y ) E n − k , q ( l x , 0 ) t n [ n ] q ! .$

On the other hand,

$I 1 = 1 2 ∑ n = 0 ∞ B n , q [ m − 1 , α ] ( 1 l , y ) t n [ n ] q ! ∑ k = 0 ∞ E k , q ( l x , 0 ) t k m k [ k ] q ! = 1 2 ∑ n = 0 ∞ ∑ k = 0 n [ n k ] q l k − n B k , q [ m − 1 , α ] ( 1 l , y ) E n − k , q ( l x , 0 ) t n [ n ] q ! .$

Therefore

$∑ n = 0 ∞ B n , q [ m − 1 , α ] ( x , y ) t n [ n ] q ! = 1 2 ∑ n = 0 ∞ ∑ k = 0 n [ n k ] q l k − n [ B k , q [ m − 1 , α ] ( 0 , y ) + B k , q [ m − 1 , α ] ( 1 l , y ) ] E n − k , q ( l x , 0 ) t n [ n ] q ! .$

□

Next we discuss some special cases of Theorem 11.

Theorem 12 The relationship

$B n , q [ m − 1 , α ] ( x , y ) = 1 2 ∑ k = 0 n [ n k ] q [ B k , q [ m − 1 , α ] ( 0 , y ) + ∑ k = 0 min ( n , m − 1 ) [ n k ] q B n − k , q [ m − 1 , α ] ( 0 , y ) + [ k ] q ∑ j = 0 k − 1 [ k − 1 j ] q B j , q [ m − 1 , α ] ( 0 , y ) B k − 1 − j , q [ 0 , − 1 ] ] E n − k , q ( x , 0 )$

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

Remark 13 Taking $q→ 1 −$ in Theorem 12, we obtain the Srivastava-Pintér addition theorem for the generalized Bernoulli and Euler polynomials.

$B n [ m − 1 , α ] ( x + y ) = 1 2 ∑ k = 0 n ( n k ) [ B k [ m − 1 , α ] ( y ) + ∑ k = 0 min ( n , m − 1 ) ( n k ) B n − k [ m − 1 , α ] ( y ) + k ∑ j = 0 k − 1 ( k − 1 j ) B j [ m − 1 , α ] ( y ) B k − 1 − j [ 0 , − 1 ] ] E n − k ( x ) .$
(11)

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

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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.

Author information

Correspondence to Nazim I Mahmudov.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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