Some identities of type 2 q-Bernoulli polynomials

Abstract

Recently, symmetric properties of some special polynomials arising from p-adic q-integrals on ${\mathbb{Z}}_{p}$ have been investigated extensively by many researchers.

In this paper, we find some symmetric identities for type 2 q-Bernoulli polynomials under the symmetry group $S_{n}$ of degree n by using the bosonic p-adic q-integral on ${\mathbb{Z}}_{p}$.

1 Introduction

In the book Ars Conjectandi, Jakob Bernoulli introduced the Bernoulli numbers for studying of the sums of powers of consecutive integers $1^{k} +2^{k} +\cdots +n^{2}$. As a generalization of these numbers, we have the Bernoulli polynomials, which are defined by the generating function to be

$$ \frac{t}{e^{t}-1}e^{xt}=\sum_{n=0} ^{\infty } B_{n}(x) \frac{t^{n}}{n!},\quad {\text{where }} \vert t \vert < 2 \pi \text{ (see [2, 4, 5, 7, 12, 15, 16, 28, 34, 35, 41])}. $$

In the special case $x=0$, $B_{n}(0)=B_{n}$ are the Bernoulli numbers. It is well known that

$$ 0^{k} +1^{k} + 2^{k}+\cdots n^{k}= \frac{1}{k+1} \bigl(B_{k+1}(n+1)-B_{k+1} \bigr), $$

for each nonnegative integer k and positive integer n (see [2, 15, 21]).

The Bernoulli numbers and polynomials have very important roles in the pure and applied mathematics. In [14], Kim and Kim represented several trigonometric functions as a formal power series involving either Bernoulli or Euler numbers, and relationships between the Bernoulli numbers and zeta functions are investigated by Arakawa et al. in [2]. In [1], authors found some applications of the Bernoulli numbers and polynomials corresponding to zeta functions, and Kim and Kim studied some special polynomials in connection with Bernoulli numbers and polynomials (see [27]).

In addition, the Bernoulli numbers and polynomials have been generalized by many researchers. For example, Carlitz degenerated the Bernoulli polynomials and investigated the properties of those polynomials in [5]. Kim represented the q-Bernoulli polynomials and numbers by using q-Volkenborn integration (see [19, 20]). In [40], authors constructed a family of modified p-adic twisted functions and gave some applications and examples related to these polynomials, and Simsek defined some new sequences containing the Bernoulli numbers and the Euler numbers and found some computation formulas and identities for those sequences (see [38]). Gaboury et al. defined some new classes of Bernoulli polynomials and found explicit formulas of those polynomials in [7], and Park and Rim defined modified q-Bernoulli polynomials with weight (see [34]). Moreover, Simsek gave the generating functions of the twisted Bernoulli polynomials and obtained some equations which are general versions of Eq. (1) in [36], and Kim et al. defined type 2 q-Bernoulli polynomials and investigated the properties of those polynomials (see [15]).

As a new generalization of the Bernoulli numbers, Jang and Kim introduced the type 2 degenerate Bernoulli polynomials in [9] and investigated relationships between some special functions and those polynomials. In addition, they found some interesting identities for those numbers and polynomials.

Carlitz generalized the Bernoulli polynomials to q-Bernoulli numbers in [3, 4, 6, 15, 21, 28, 34, 35] as follows:

$$ \beta _{0,q}=1,\qquad q(q\beta _{q}+1)^{n}-\beta _{n,q}= \textstyle\begin{cases} 1,&\text{if }n=1, \\ 0,&\text{if }n>1, \end{cases} $$

with the usual convection about replacing $\beta _{q} ^{n}$ by $\beta _{n,q}$, and Kim defined the modified q-Bernoulli numbers as follows:

$$ B_{0,q}=\frac{q-1}{\log q},\qquad (qB_{q}+1 )^{n}-B_{n,q}= \textstyle\begin{cases} 1,&{\text{if }}n=1, \\ 0, & {\text{if }}n>1, \end{cases} $$

(1)

with the usual convection about replacing $B_{q} ^{n}$ by $B_{n,q}$ (see [18, 21, 22]).

For a given prime number p, $\mathbb{Z}_{p}$, $\mathbb{Q}_{p}$, and $\mathbb{C}_{p}$ denote the ring of p-adic integers, the field of p-adic rational numbers, and the completions of algebraic closure of $\mathbb{Q}_{p}$, respectively. The p-adic norm is normalized as $|p|_{p} = \frac{1}{p}$.

Let $q \in {\mathbb{C}}_{p}$ be an indeterminate with $|q-1|_{p} < p^{- \frac{1}{p-1}}$. Then the q-analogue of numberx are defined as $[x]_{q} =\frac{1- q^{x}}{1-q}$. Note that $\lim_{q \rightarrow 1} [x]_{q} = x$ for each $x \in \mathbb{Z}_{p}$.

By (1), we know that the power sums of consecutive nonnegative q-integers are given by

$$ [0]_{q} ^{k} +q[1]_{q} ^{k}+q^{2}[2]_{q} ^{k} +\cdots +q^{n-1}[n-1]_{q} ^{k} = \frac{1}{k+1} \bigl(B_{k+1,q}(n)-B_{k+1,q} \bigr)\quad (n\geq 1, k \geq 0) $$

(see [21, 23, 31, 39]).

For the set of all uniformly differentiable on ${\mathbb{Z}}_{p}$ denoted by $\operatorname{UD}(\mathbb{Z}_{p})$, the bosonicp-adicq-integrals on${\mathbb{Z}}_{p}$ are defined by the Kim as follows:

$$ \int _{\mathbb{Z}_{p}}f(x)\,d\mu _{q}(x)=\lim _{N \rightarrow \infty } \frac{1}{[p^{N}]_{q}}\sum_{x=0}^{p^{N}-1} f(x) q^{x}\quad \text{(see [18--20])}. $$

(2)

In [9], the authors defined type 2 Bernoulli polynomials as follows:

$$ \sum_{n=0} ^{\infty } b_{n}(x) \frac{t^{n}}{n!}= \frac{t}{e^{t}-e^{-t}}e^{xt}, $$

and found relations between some special functions or numbers and those polynomials (see [9, 12, 15, 27]). Note that a Witt-type formula of the type 2 Bernoulli polynomials is

$$ b_{n}(x)= { \int _{\mathbb{Z}_{p}}}(2y+x+1)^{n} \,d\mu _{0}(y) \quad \text{(see [9, 12])}, $$

for each nonnegative integer n where

$$ { \int _{\mathbb{Z}_{p}}}f(x) \,d\mu _{0}(x)=\lim _{N\rightarrow \infty } \frac{1}{p^{N}} \sum_{x=0} ^{p^{N}-1} f(x) $$

is the bosonic p-adic integral on ${\mathbb{Z}}_{p}$.

As a natural generalization of the type 2 Bernoulli polynomials, Kim et al. defined the type 2 q-Bernoulli polynomials by using the bosonic p-adic q-integral on ${\mathbb{Z}}_{p}$ as follows:

$$ \sum_{n=0} ^{\infty } b_{n,q}(x) \frac{t^{n}}{n!}={ \int _{\mathbb{Z}_{p}}}q^{-y} e^{[2y+x+1]_{q}t}\,d\mu _{q}(y)\quad \text{(see [15, 28])}, $$

and found the power sums of consecutive odd positive q-integers are represented by the type 2 q-Bernoulli polynomials as follows:

$$ q[1]_{q} ^{k} +q^{3} [3]_{q} ^{k} +\cdots +q^{2n-1}[2n-1]_{q} ^{k}= \frac{1}{2(k+1)} \bigl(b_{k+1,q}(2n)-b_{k+1,q} \bigr), $$

for each positive integer n and each nonnegative integer k (see [12, Eq. (2.13), Theorem 2.2]).

In the past decade, many symmetric identities of special functions have been found by many authors (see [6, 8, 10–13, 17, 24–26, 28–30, 32, 33, 37]). In particular, Kim et al. [28] obtained some symmetric identities for type 2 q-Bernoulli polynomials under symmetry group $S_{3}$, and Duran et al. [6] investigated some symmetric identities of Carlitz’s generalized twisted q-Bernoulli polynomials under the symmetric group of order 3. He et al. [8] studied some symmetric identities on a sequence of polynomials and derived interesting identities involving Bernoulli numbers and polynomials as particular cases of those identities.

In this paper, we derive symmetric identities for type 2 q-Bernoulli polynomials under the symmetry group $S_{n}$ of degree n by using the bosonic p-adic q-integral on ${\mathbb{Z}}_{p}$, and the proof methods, which are found by Kim, are also used as good tools in this paper(see [13, 25, 29, 30]). In particular, those identities are generalizations of the results of Kim et al. [28].

2 Symmetric identities for the type 2 q-Bernoulli polynomials

Let $t \in {\mathbb{C}}_{p}$ with $|t|_{p}< p^{-\frac{1}{p-1}}$, and let $S_{n}$ be the symmetry group of degree n. For each positive integers $w_{1},\ldots ,w_{n}$ and each nonnegative integers $k_{1},\ldots ,k_{n-1}$, we consider the following integral equation for the bosonic p-adic q-integral on ${\mathbb{Z}}_{p}$;

$$ \begin{aligned}[b] &{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}w_{2}\cdots w_{n-1}y}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )y+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t}\,d\mu _{q^{w_{1}w_{2} \cdots w_{n-1}}}(y) \\ &\quad =\lim_{N\rightarrow \infty } \frac{1}{[p^{N}]_{q^{w_{1}\cdots w_{n-1}}}}\sum _{y=0} ^{p^{N}-1}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )y+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t} \\ &\quad =\lim_{N\rightarrow \infty } \frac{1}{[w_{n}p^{N}]_{q^{w_{1}\cdots w_{n-1}}}}\sum _{y=0} ^{w_{n}p^{N}-1}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )y+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t} \\ &\quad =\lim_{N\rightarrow \infty } \frac{1}{[w_{n}p^{N}]_{q^{w_{1}\cdots w_{n-1}}}} \sum _{l=0} ^{w_{n}-1} \sum_{y=0} ^{p^{N}-1}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )(l+w_{n}y)+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n} \sum _{i=1} ^{n-1} (\prod _{\substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t} \\ &\quad = [w_{1}\cdots w_{n-1} ]_{q}\lim _{N\rightarrow \infty } \frac{1}{[w_{1}\cdots w_{n}p^{N}]_{q}} \\ &\qquad{}\times \sum_{l=0} ^{w_{n}-1} \sum _{y=0} ^{p^{N}-1}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )(l+w_{n}y)+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t}. \end{aligned} $$

(3)

By (3) we have

$$ \begin{aligned}[b] &\frac{1}{ [w_{1}\cdots w_{n-1} ]_{q}} \\ & \qquad{}\times \prod _{m=1} ^{n-1}\sum_{k_{m}=0} ^{w_{m}-1}{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}\cdots w_{n-1}y}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )y+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t}\,d\mu _{q^{w_{1} \cdots w_{n-1}}}(y) \\ &\quad =\lim_{N\rightarrow \infty } \frac{1}{ [w_{1}\cdots w_{n}p^{N} ]_{q}}\\ &\qquad {}\times\prod _{m=1} ^{n} \sum_{k_{m}=0} ^{w_{m}-1} \sum_{y=0} ^{p^{N}-1}e^{ [2 ( \prod _{i=1} ^{n-1}w_{i} )(k_{m}+w_{n}y)+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t}. \end{aligned} $$

(4)

If we put

$$ \begin{aligned}[b] &F(w_{1},\ldots ,w_{n})\\ &\quad = \frac{1}{ [w_{1}\cdots w_{n-1} ]_{q}} \\ &\qquad{}\times \prod_{m=1} ^{n-1} \sum_{k_{m}=0} ^{w_{m}-1}{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}\cdots w_{n-1}y}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )y+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t}\, d\mu _{q^{w_{1} \cdots w_{n-1}}}(y), \end{aligned} $$

(5)

then, by (5), we obtain the following theorem.

Theorem 2.1

Let$w_{1},w_{2},\ldots ,w_{n}$be positive integers. For each$\sigma \in S_{n}$, the$F (w_{\sigma (1)},w_{\sigma (2)},\ldots , w_{\sigma (n)} )$have the same value.

Note that, by the definition of the q-analogue of the number x,

$$ \begin{aligned}[b] & \Biggl[2 \Biggl(\prod _{i=1} ^{n-1}w_{i} \Biggr)y+ \Biggl(\prod _{i=1} ^{n}w_{i} \Biggr) (x+1)+2w_{n}\sum_{i=1} ^{n-1} \Biggl(\prod_{\substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q} \\ &\quad = \Biggl[\prod_{i=1} ^{n-1}w_{i} \Biggr]_{q} \biggl[2y+w_{n}(x+1)+2 \frac{w_{n}}{w_{1}}k_{1}+ \cdots +2\frac{w_{n}}{w_{n-1}}k_{n-1} \biggr]_{q^{w_{1} \cdots w_{n-1}}} \\ &\quad = \Biggl[\prod_{i=1} ^{n-1}w_{i} \Biggr]_{q} \Biggl[2y+w_{n}(x+1)+2 \sum _{i=1} ^{n-1} \frac{w_{n}}{w_{i}}k_{i} \Biggr]_{q^{w_{1}\cdots w_{n-1}}}. \end{aligned} $$

(6)

By the definition of type 2 q-Bernoulli polynomials and (6), we get

$$ \begin{aligned}[b] &{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}w_{2}\cdots w_{n-1}y}e^{ [2 (\prod _{i=1} ^{n-1}w_{i} )y+ (\prod _{i=1} ^{n}w_{i} )(x+1)+2w_{n}\sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} )k_{i} ]_{q}t}\,d\mu _{q^{w_{1}w_{2} \cdots w_{n-1}}}(y) \\ &\quad =\sum_{m=0} ^{\infty } \Biggl[\prod _{i=1} ^{n-1}w_{i} \Biggr]_{q} ^{m}{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}w_{2}\cdots w_{n-1}y} \Biggl[2y+w_{n}(x+1)\\ &\qquad {}+2 \sum_{i=1} ^{n-1} \frac{w_{n}}{w_{i}}k_{i}-1+1 \Biggr]_{q^{w_{1} \cdots w_{n-1}}} ^{m}\,d\mu _{q^{w_{1}w_{2}\cdots w_{n-1}}}(y) \frac{t^{m}}{m!} \\ &\quad =\sum_{m=0} ^{\infty } \Biggl[\prod _{i=1} ^{n-1}w_{i} \Biggr]_{q} ^{m} b_{m,q^{w_{1}\cdots w_{n-1}}} \Biggl(w_{n}(x+1)+2\sum _{i=1} ^{n-1} \frac{w_{n}}{w_{i}}k_{i}-1 \Biggr)\frac{t^{m}}{m!}, \end{aligned} $$

(7)

and so by (7), we know that

$$ \begin{aligned}[b] &{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}\cdots w_{n-1}y} \Biggl[2 \Biggl(\prod _{i=1} ^{n-1}w_{i} \Biggr)y+ \Biggl(\prod _{i=1} ^{n}w_{i} \Biggr) (x+1)+2w_{n}\sum_{i=1} ^{n-1} \Biggl(\prod_{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q} ^{m} \,d \mu _{q^{w_{1}\cdots w_{n-1}}}(y) \\ &\quad = \Biggl[\prod_{i=1} ^{n-1}w_{i} \Biggr]_{q} ^{m}b_{m,q^{w_{1} \cdots w_{n-1}}} \Biggl(w_{n}(x+1)+2 \sum_{i=1} ^{n-1} \frac{w_{n}}{w_{i}}k_{i}-1 \Biggr). \end{aligned} $$

(8)

By Theorem 2.1 and (8), we obtain the following theorem.

Theorem 2.2

For each nonnnegative integermand each set of positive integers$w_{1},w_{2},\ldots , w_{n}$,

$$ \prod_{m=1} ^{n-1}\sum _{k_{\sigma (m)}=0} ^{w_{\sigma (m)}-1} \Biggl[\prod _{i=1} ^{n-1}w_{\sigma (i)} \Biggr]_{q} ^{m-1} b_{m,q^{w_{ \sigma (1)}\cdots w_{{\sigma (n-1)}}}} \Biggl(w_{\sigma (n)}(x+1)+2 \sum _{i=1} ^{n-1} \frac{w_{\sigma (n)}}{w_{\sigma (i)}}k_{\sigma (i)}-1 \Biggr) $$

have the same value for any$\sigma \in S_{n}$.

By the definition of q-analogue of number x, we know that

$$ \begin{aligned}[b] & \Biggl[2y+w_{n}(x+1)+2w_{n} \sum_{i=1} ^{n-1} \frac{k_{i}}{w_{i}} \Biggr]_{q^{w_{1}w_{2}\cdots w_{n-1}}} \\ &\quad =\frac{[w_{n}]_{q}}{ [\prod_{i=1} ^{n-1}w_{i} ]_{q}} \Biggl[2\sum_{i=1} ^{n-1} \Biggl(\prod_{\substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q^{w_{n}}}\\ &\qquad {}+q^{2w_{n}\sum _{i=1} ^{n-1} ( \prod _{\substack{j=1\\j\neq i}} ^{n-1} w_{j} )k_{i}} \bigl[2y+w_{n}(x+1) \bigr]_{q^{w_{1}w_{2}\cdots w_{n-1}}}, \end{aligned} $$

(9)

and by (9), we get

$$ \begin{aligned}[b] &{ \int _{\mathbb{Z}_{p}}}q^{-w_{1}\cdots w_{n-1}y} \Biggl[2y+w_{n}(x+1)+2w_{n} \sum_{i=1} ^{n-1}\frac{k_{i}}{w_{i}} \Biggr]_{q^{w_{1}w_{2}\cdots w_{n-1}}} ^{m} \,d\mu _{q^{w_{1}\cdots w_{n-1}}}(y) \\ &\quad =\sum_{l=0} ^{m} \binom{m}{l} \frac{[w_{n}]_{q} ^{m-l} }{ [\prod_{i=1} ^{n-1} w_{i} ]_{q} ^{m-l}} \Biggl[2\sum_{i=1} ^{n-1} \Biggl(\prod_{\substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q^{w_{n}}} ^{m-l} q^{2w_{n} l \sum _{i=1} ^{n-1} (\prod _{\substack{j=1\\j\neq i}} ^{n-1} w_{j} )k_{i}} \\ &\qquad{}\times { \int _{\mathbb{Z}_{p}}}q^{-w_{1}\cdots w_{n-1}y} \bigl[2y+w_{n}(x+1)-1+1 \bigr]_{q^{w_{1}w_{2}\cdots w_{n-1}}} ^{l} \,d\mu _{q^{w_{1}\cdots w_{n-1}}}(y) \\ &\quad =\sum_{l=0} ^{m} \binom{m}{l} \frac{[w_{n}]_{q} ^{m-l} }{ [\prod_{i=1} ^{n-1} w_{i} ]_{q} ^{m-l}} \Biggl[2\sum_{i=1} ^{n-1} \Biggl(\prod_{\substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q^{w_{n}}} ^{m-l} \\ &\qquad{}\times q^{2w_{n} l \sum _{i=1} ^{n-1} (\prod _{\substack{j=1\\j\neq i}} ^{n-1} w_{j} )k_{i}} b_{l,q^{w_{1}\cdots w_{n-1}}} (w_{n}x+w_{n}-1 ). \end{aligned} $$

(10)

By Theorem 2.1 and (10), we have

$$ \begin{aligned}[b] &\prod _{m=1} ^{n-1}\sum_{k_{m}=0} ^{w_{m}-1} \Biggl[ \prod_{i=1} ^{n-1}w_{i} \Biggr]_{q} ^{m-1} { \int _{\mathbb{Z}_{p}}}q^{-w_{1} \cdots w_{n-1}y} \Biggl[2y+w_{n}(x+1)\\ &\qquad {}+2w_{n} \sum_{i=1} ^{n-1} \frac{k_{i}}{w_{i}} \Biggr]_{q^{w_{1}w_{2}\cdots w_{n-1}}} ^{m} \,d\mu _{q^{w_{1} \cdots w_{n-1}}}(y) \\ &\quad =\prod_{m=1} ^{n-1}\sum _{k_{m}=0} ^{w_{m}-1} \sum_{l=0} ^{m} \Biggl[\prod_{r=1} ^{n-1}w_{r} \Biggr]_{q} ^{l-1} \binom{m}{l} [w_{n}]_{q} ^{m-l} \Biggl[2\sum _{i=1} ^{n-1} \Biggl(\prod _{ \substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q^{w_{n}}} ^{m-l} q^{2w_{n} l \sum _{i=1} ^{n-1} (\prod _{\substack{j=1\\j\neq i}} ^{n-1} w_{j} )k_{i}} \\ &\qquad{}\times b_{l,q^{w_{1}\cdots w_{n-1}}} (w_{n}x+w_{n}-1 ) \\ &\quad =\sum_{l=0} ^{m} \Biggl[\prod _{r=1} ^{n-1}w_{r} \Biggr]_{q} ^{l-1} \binom{m}{l} [w_{n}]_{q} ^{m-l} R_{m,q^{w_{n}}} ( w_{1}, \ldots ,w_{n-1}|2w_{n}l )b_{l,q^{w_{1}\cdots w_{n-1}}} (w_{n}x+w_{n}-1 ), \end{aligned} $$

(11)

where

$$ \begin{aligned}[b] R_{m,q} ( w_{1},\ldots ,w_{n-1}| l )=\prod_{m=1} ^{n-1}\sum _{k_{m}=0} ^{w_{m}-1} \Biggl[2\sum _{i=1} ^{n-1} \Biggl(\prod_{\substack{j=1\\j\neq i}} ^{n-1}w_{j} \Biggr)k_{i} \Biggr]_{q} ^{m-l} q^{l \sum _{i=1} ^{n-1} (\prod _{ \substack{j=1\\j\neq i}} ^{n-1} w_{j} )k_{i}}. \end{aligned} $$

Hence, by (11), we obtain the following theorem.

Theorem 2.3

For each set of positive integers$w_{1},\ldots ,w_{n}$and each nonnegative integerm,

$$ \begin{aligned}[b] &\sum_{l=0} ^{m} \Biggl[\prod_{r=1} ^{n-1}w_{\sigma (r)} \Biggr]_{q} ^{l-1} \binom{m}{l} [w_{\sigma (n)}]_{q} ^{m-l} R_{m,q^{w_{ \sigma (n)}}} ( w_{\sigma (1)},\ldots ,w_{\sigma (n-1)} |2w_{\sigma (n)}l ) \\ &\quad{}\times b_{l,q^{w_{\sigma (1)}\cdots w_{\sigma (n-1)}}} (w_{ \sigma (n)}x+w_{\sigma (n)}-1 ) \end{aligned} $$

have the same value for any$\sigma \in S_{n}$.

3 Conclusions

The Bernoulli numbers and polynomials are very important things in the pure and applied mathematics, and closely connected with the Stirling numbers of the first and second kind, harmonic numbers, the Riemann zeta function, and so on. In addition, the Bernoulli numbers and polynomials can represent the power sums of consecutive integers.

In [15], Kim et al. showed that power sums of consecutive positive odd q-integers can be expressed by means of type 2 q-Bernoulli polynomials as follows:

$$ \sum_{l=0} ^{n-1} q^{2l+1} [2l+1]_{q} ^{m} =\frac{1}{m+1} \bigl(b_{m+1,q}(2n)-b_{m+1,q} \bigr). $$

In this paper, we study that the function $F (w_{\sigma (1)},w_{\sigma (2)},\ldots ,w_{\sigma (n)} )$ about the type 2 q-Bernoulli polynomials is invariant for any $\sigma \in S_{n}$ where $S_{n}$ is the symmetry group of degree n. By the those invariance, we derive some symmetric identities for the type 2 q-Bernoulli polynomials arising from the bosonic p-adic q-integral on ${\mathbb{Z}}_{p}$.

If we put $n=3$ or $w_{4}=w_{5}=\cdots =w_{n}=1$, then Theorems 2.2 and 2.3 are the results of Kim et al. [28]. In addition, if we put $w_{2}=w_{3}=\cdots =w_{n}=1$, then we obtain the equation

$$ b_{n,q}\bigl(w_{1}(x+1)-1\bigr)= [w_{1}]_{q} ^{n-1} \sum_{j=0} ^{w_{1}-1}b_{n,q^{w_{1}}} \biggl(x+ \frac{1}{w_{1}}k_{1} \biggr) $$

which is the same result as in [28].