New identities and relations derived from the generalized Bernoulli polynomials, Euler polynomials and Genocchi polynomials

Abstract

In this article, we give some identities for the q-Bernoulli polynomials, q-Euler polynomials and q-Genocchi polynomials and recurrence relations between these polynomials in (Mahmudov in Discrete Dyn. Nat. Soc. 2012:169348, 2012; Mahmudov in Adv. Differ. Equ. 2013:1, 2013).

MSC:05A10, 11B65, 28B99, 11B68.

1 Introduction, definitions and notations

In the usual notations, let $B_n(x)$ and $E_n(x)$ denote, respectively, the classical Bernoulli and Euler polynomials of degree n in x, defined by the generating functions

$$\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{t^n}{n!}=\frac{t}{e^t-1}{e}^{xt},|t|<2\pi$$

and

$$\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!}=\frac{2}{e^t+1}{e}^{xt},|t|<\pi .$$

Also, let

$$B_n:=B_n(0)\text{and}{E}_n:=E_n(0),$$

where $B_n$ and $E_n$ are, respectively, the Bernoulli and Euler numbers of order n.

Carlitz first extended the classical Bernoulli polynomials and numbers, Euler polynomials and numbers [1]. There are numerous recent investigations on this subject by many authors. Cheon [2], Kurt [3], Luo [4], Luo and Srivastava [5], Srivastava et al. [6, 7], Tremblay et al. [8], and Mahmudov [9, 10].

Throughout this paper, we always make use of the following notation: ℕ denotes the set of natural numbers and ℂ denotes the set of complex numbers.

The q-numbers and q-factorial are defined by

$${[a]}_q=\frac{1-q^a}{1-q},q\ne 1,{[n]}_q!={[n]}_q{[n-1]}_q\cdots {[2]}_q{[1]}_q,n\in \mathbb{N},a\in \mathbb{C},$$

respectively, where ${[0]}_q!=1$, $n\in \mathbb{N}$, $a\in \mathbb{C}$. The q-polynomials coefficient is defined by

$${\left[\begin{array}{c}n\\ k\end{array}\right]}_q=\frac{{(q:q)}_n}{{(q:q)}_{n-k}{(q:q)}_k},$$

where ${(q:q)}_n=(1-q)\cdots {(1-q^n)}_n$.

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

$${(x+y)}_q^n=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{\frac{k(k-1)}{2}}{x}^{n-k}{y}^k.$$

The q-binomial formula is known as

$${(n:q)}_a={(1-a)}_q^n=\prod _{j=0}^{n-1}(1-q^{j}a)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{\frac{k(k-1)}{2}}{(-1)}^k{a}^k.$$

The q-exponential functions are given by

$$e_q(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{[n]}_q!}=\prod _{k=0}^{\mathrm{\infty }}\frac{1}{(1-(1-q)q^{k}z)},0<|q|<1,|z|<\frac{1}{|1-q|}$$

and

$$E_q(z)=\sum _{n=0}^{\mathrm{\infty }}{q}^{\frac{n(n-1)}{2}}\frac{z^n}{{[n]}_q!}=\prod _{k=0}^{\mathrm{\infty }}(1+(1-q)q^{k}z),0<|q|<1,z\in \mathbb{C}.$$

From these forms, 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

$$D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z},0<|q|<1,0\ne z\in \mathbb{C}.$$

The above q-standard notation can be found in [10].

Mahmudov defined and studied properties of the following generalized q-Bernoulli polynomials ${\mathcal{B}}_{n,q}^{(\alpha )}(x,y)$ of order α and q-Euler polynomials ${\mathcal{E}}_{n,q}^{(\alpha )}(x,y)$ of order α as follows [10].

Let $q\in \mathbb{C}$, $\alpha \in \mathbb{N}$ and $0<|q|<1$. The q-Bernoulli numbers ${\mathcal{B}}_{n,q}^{(\alpha )}$ and polynomials ${\mathcal{B}}_{n,q}^{(\alpha )}(x,y)$ in x, y of order α are defined by means of the generating functions

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}^{(\alpha )}\frac{t^n}{{[n]}_q!}={\left(\frac{t}{e_q(t)-1}\right)}^{\alpha },|t|<2\pi ,$$

(1)

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}^{(\alpha )}(x,y)\frac{t^n}{{[n]}_q!}={\left(\frac{t}{e_q(t)-1}\right)}^{\alpha }{e}_q(tx)E_q(ty),|t|<2\pi .$$

(2)

The q-Euler numbers ${\mathcal{E}}_{n,q}^{(\alpha )}$ and polynomials ${\mathcal{E}}_{n,q}^{(\alpha )}(x,y)$ in x, y of order α are defined by means of the generating functions

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_{n,q}^{(\alpha )}\frac{t^n}{{[n]}_q!}={\left(\frac{2}{e_q(t)+1}\right)}^{\alpha },|t|<\pi ,$$

(3)

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_{n,q}^{(\alpha )}(x,y)\frac{t^n}{{[n]}_q!}={\left(\frac{2}{e_q(t)+1}\right)}^{\alpha }{e}_q(tx)E_q(ty),|t|<\pi .$$

(4)

The q-Genocchi numbers ${\mathcal{G}}_{n,q}^{(\alpha )}$ and polynomials ${\mathcal{G}}_{n,q}^{(\alpha )}(x,y)$ in x, y of order α are defined by means of the generating functions

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}\frac{t^n}{{[n]}_q!}={\left(\frac{2t}{e_q(t)+1}\right)}^{\alpha },|t|<\pi ,$$

(5)

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)\frac{t^n}{{[n]}_q!}={\left(\frac{2t}{e_q(t)+1}\right)}^{\alpha }{e}_q(tx)E_q(ty),|t|<\pi .$$

(6)

It is obvious that

$$\begin{array}{c}{\mathcal{B}}_{n,q}^{(\alpha )}={\mathcal{B}}_{n,q}^{(\alpha )}(0,0),\underset{q\to 1^{-}}{lim}{\mathcal{B}}_{n,q}^{(\alpha )}(x,y)={\mathcal{B}}_n^{(\alpha )}(x+y),\underset{q\to 1^{-}}{lim}{\mathcal{B}}_{n,q}^{(\alpha )}={\mathcal{B}}_n^{(\alpha )},\hfill \\ {\mathcal{E}}_{n,q}^{(\alpha )}={\mathcal{E}}_{n,q}^{(\alpha )}(0,0),\underset{q\to 1^{-}}{lim}{\mathcal{E}}_{n,q}^{(\alpha )}(x,y)={\mathcal{E}}_n^{(\alpha )}(x+y),\underset{q\to 1^{-}}{lim}{\mathcal{E}}_{n,q}^{(\alpha )}={\mathcal{E}}_n^{(\alpha )}\hfill \end{array}$$

and

$${\mathcal{G}}_{n,q}^{(\alpha )}={\mathcal{G}}_{n,q}^{(\alpha )}(0,0),\underset{q\to 1^{-}}{lim}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)={\mathcal{G}}_n^{(\alpha )}(x+y),\underset{q\to 1^{-}}{lim}{\mathcal{G}}_{n,q}^{(\alpha )}={\mathcal{G}}_n^{(\alpha )}.$$

From (2), (4) and (6), it is easy to check that

$$\begin{array}{c}{\mathcal{B}}_{n,q}^{(\alpha )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{B}}_{n-k,q}(x,0){\mathcal{B}}_{k,q}^{(\alpha -1)}(0,y),\hfill \\ {\mathcal{E}}_{n,q}^{(\alpha )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{E}}_{n-k,q}(x,0){\mathcal{E}}_{k,q}^{(\alpha -1)}(0,y)\hfill \end{array}$$

and

$${\mathcal{G}}_{n,q}^{(\alpha )}(x,y)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{G}}_{n-k,q}(x,0){\mathcal{G}}_{k,q}^{(\alpha -1)}(0,y).$$

In this work, we give a different form of the analogue of the Srivastava-Pintér addition theorem.

More precisely, we prove

$$\begin{array}{c}{\mathcal{G}}_{n,q}(x,y)=y{\mathcal{G}}_{n-1,q}(x,qy)+x{\mathcal{G}}_{n-1,q}(x,y)\hfill \\ \phantom{{\mathcal{G}}_{n,q}(x,y)=}+\frac{1}{{[n]}_q}\{{\mathcal{G}}_{n,q}(x,y)-\frac{1}{2}\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{G}}_{k,q}(x,y){\mathcal{G}}_{n-k,q}(1,0)\},\hfill \\ \sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{G}}_{k,q}(x,y)+{\mathcal{G}}_{n,q}(x,y)=2{[n]}_q{(x+y)}_q^{n-1},\hfill \\ {\mathcal{G}}_{n,q}^{(\alpha )}(x,y)\hfill \\ =\frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(x,0)m^{j-k}+{\mathcal{G}}_{k,q}^{(\alpha )}(x,0)\}{\mathcal{G}}_{n+1-k,q}(0,my)m^{k-n}\hfill \\ =\frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(0,y)m^{j-k}+{\mathcal{G}}_{k+1,q}^{(\alpha )}(0,y)\}\hfill \\ \times {\mathcal{G}}_{n+1-k,q}(mx,0)m^{k-n},\hfill \\ {\mathcal{G}}_{n,q}^{(\alpha )}(x,y)\hfill \\ =\frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(x,0)m^{j-n}-{\mathcal{G}}_{k,q}^{(\alpha )}(x,0)\}{\mathcal{B}}_{n+1-k,q}(0,my)m^{k-n},\hfill \\ {\mathcal{B}}_{n,q}^{(\alpha )}(x,y)\hfill \\ =\frac{1}{2}\sum _{r=0}^{n+1}{\left[\begin{array}{c}n+1\\ r\end{array}\right]}_q\frac{1}{{[n+1]}_q}(\sum _{r=0}^k{\left[\begin{array}{c}k\\ r\end{array}\right]}_q{\mathcal{B}}_{k,q}^{(\alpha )}(x,0)m^{k-r}+{\mathcal{B}}_{r,q}^{(\alpha )}(x,0))\hfill \\ \times {\mathcal{G}}_{n+1-r,q}(0,my)m^{r-n}.\hfill \end{array}$$

2 Main theorems

Proposition 2.1 The generalized q-Bernoulli and q-Euler polynomials satisfy the following relations:

$$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{B}}_{k,q}^{(\alpha )}(x,0){\mathcal{B}}_{n-k,q}^{(-\alpha )}=x^n,$$

(7)

$$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{B}}_{k,q}^{(\alpha )}(0,y){\mathcal{B}}_{n-k,q}^{(-\alpha )}=q^{\frac{n(n-1)}{2}}{y}^n,$$

(8)

$${\mathcal{B}}_{n,q}^{(\alpha )}(x,y)=\sum _{l=0}^n{\left[\begin{array}{c}n\\ l\end{array}\right]}_q{\mathcal{B}}_{n-l,q}^{(\alpha )}(0,y)\sum _{k=0}^l{\left[\begin{array}{c}l\\ k\end{array}\right]}_q{\mathcal{E}}_{k,q}^{(\alpha )}(x,0){\mathcal{E}}_{l-k,q}^{(-\alpha )}(0,0),$$

(9)

$${\mathcal{E}}_{n,q}^{(\alpha )}(x,y)=\sum _{l=0}^n{\left[\begin{array}{c}n\\ l\end{array}\right]}_q{\mathcal{E}}_{n-l,q}^{(\alpha )}(0,y)\sum _{k=0}^l{\left[\begin{array}{c}l\\ k\end{array}\right]}_q{\mathcal{E}}_{k,q}^{(\alpha )}(x,0){\mathcal{B}}_{l-k,q}^{(-\alpha )}(0,0).$$

(10)

Proposition 2.2 For $x,y,z\in \mathbb{C}$, the following relations hold true:

$${\mathcal{G}}_{n,q}^{(\alpha )}(x+z,y)=\sum _{p=0}^n{\left[\begin{array}{c}n\\ p\end{array}\right]}_q{\mathcal{G}}_{n-p,q}^{(\alpha )}(0,y)\sum _{r=0}^p{\left[\begin{array}{c}p\\ r\end{array}\right]}_q{x}^r{z}^{p-r},$$

(11)

$$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{G}}_{k,q}^{(\alpha )}(x,y){\mathcal{G}}_{n-k,q}^{(-\alpha )}(0,0)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{x}^k{y}^{n-k}{q}^{\frac{(n-k)(n-k-1)}{2}}={(x+y)}_q^n.$$

(12)

Proof The proof of these propositions can be found from (1)-(6). □

Theorem 2.3 The generalized q-Genocchi polynomials satisfy the following recurrence relation:

$$\begin{array}{rcl}{\mathcal{G}}_{n,q}(x,y)& =& y{\mathcal{G}}_{n-1,q}(x,qy)+x{\mathcal{G}}_{n-1,q}(x,y)\\ +\frac{1}{{[n]}_q}\{{\mathcal{G}}_{n,q}(x,y)-\frac{1}{2}\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{G}}_{k,q}(x,y){\mathcal{G}}_{n-k,q}(1,0)\}.\end{array}$$

(13)

Proof In (6) for $\alpha =1$, we take the q-derivative of the generalized q-Genocchi polynomials ${\mathcal{G}}_{n,q}(x,y)$ according to t. We note that

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{D}_{q,t}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}& =& D_{q,t}\{\frac{2t}{e_q(t)+1}{e}_q(tx)E_q(yt)\}\\ =& \frac{2e_q(tx)E_q(yt)}{e_q(t)+1}+\frac{y2t{e}_q(tx)E_q(yt)}{e_q(t)+1}+\frac{x2t{e}_q(tx)E_q(yt)}{e_q(t)+1}\\ -\frac{2t{e}_q(tx)E_q(yt)}{e_q(t)+1}\frac{e_q(x)}{e_q(t)+1}\end{array}$$

and

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n+1,q}(x,y)\frac{t^n}{{[n]}_q!}& =& \frac{1}{t}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}+y\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,qy)\frac{t^n}{{[n]}_q!}\\ +x\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}-\frac{1}{2t}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(1,0)\frac{t^n}{{[n]}_q!}.\end{array}$$

If we take necessary operation, comparing the coefficients of $\frac{t^n}{{[n]}_q!}$, we have (13). □

Theorem 2.4 There is the following relation for the q-Genocchi polynomials:

$$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q({\mathcal{G}}_{k,q}^{(\alpha )}(x,0)+{\mathcal{G}}_{k,q}^{(\alpha )}(x,-1))=2{[n]}_q{\mathcal{G}}_{n-1,q}^{(\alpha -1)}(x,0).$$

(14)

Proof From (6) and $e_q(z)E_q(-z)=1$, we have

$$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}+\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,-1)\frac{t^n}{{[n]}_q!}={\left(\frac{2t}{e_q(t)+1}\right)}^{\alpha }{e}_q(tx)(1+E_q(-t))$$

and

$$\sum _{n=0}^{\mathrm{\infty }}({\mathcal{G}}_{n,q}^{(\alpha )}(x,0)+{\mathcal{G}}_{n,q}^{(\alpha )}(x,-1))\frac{t^n}{{[n]}_q!}=2t\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha -1)}(x,0)\frac{t^n}{{[n]}_q!}.$$

Thus, we obtain

$$\sum _{n=0}^{\mathrm{\infty }}\left\{\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q({\mathcal{G}}_{k,q}^{(\alpha )}(x,0)+{\mathcal{G}}_{k,q}^{(\alpha )}(x,-1))\right\}\frac{t^n}{{[n]}_q!}=2\sum _{n=1}^{\mathrm{\infty }}{[n]}_q{\mathcal{G}}_{n-1,q}^{(\alpha -1)}(x,0)\frac{t^n}{{[n]}_q!}.$$

From this last equality, we have (14). □

Theorem 2.5 There is the following identity for the q-Genocchi polynomials:

$$\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathcal{G}}_{k,q}(x,y)+{\mathcal{G}}_{n,q}(x,y)=2{[n]}_q{(x+y)}_q^{n-1}.$$

(15)

Proof From $e_q(t)E_q(-t)=1$, we write as

$$\begin{array}{c}\frac{1}{E_q(-t)+1}=1-\frac{1}{e_q(t)+1},\hfill \\ \frac{2t{e}_q(tx)E_q(yt)}{E_q(-t)+1}=2t{e}_q(tx)E_q(yt)-2t\frac{e_q(tx)E_q(yt)}{e_q(t)+1},\hfill \\ \frac{2t}{e_q(t)+1}{e}_q(tx)E_q(yt)e_q(t)=2t{e}_q(tx)E_q(ty)-\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!},\hfill \\ \sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{{[n]}_q!}=2\sum _{n=0}^{\mathrm{\infty }}{(x,y)}_q^n\frac{t^{n+1}}{{[n]}_q!}-\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(x,y)\frac{t^n}{{[n]}_q!}.\hfill \end{array}$$

By using the Cauchy product, compression of the results, we have (15). □

Theorem 2.6 There are the following relationships for the q-Genocchi polynomials:

$$\begin{array}{rcl}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)& =& \frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(x,0)m^{j-k}+{\mathcal{G}}_{k,q}^{(\alpha )}(x,0)\}\\ \times {\mathcal{G}}_{n+1-k,q}(0,my)m^{k-n},\end{array}$$

(16)

$$\begin{array}{rcl}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)& =& \frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(0,y)m^{j-k}+{\mathcal{G}}_{k+1,q}^{(\alpha )}(0,y)\}\\ \times {\mathcal{G}}_{n+1-k,q}(mx,0)m^{k-n}.\end{array}$$

(17)

Proof Proof of (16), we write

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)\frac{t^n}{{[n]}_q!}& =& {\left(\frac{2t}{e_q(t)+1}\right)}^{\alpha }{e}_q(tx)E_q(ty)\\ =& {\left(\frac{2t}{e_q(t)+1}\right)}^{\alpha }{e}_q(tx)\frac{e_q(\frac{t}{m})+1}{\frac{t}{m}}\frac{\frac{t}{m}}{e_q(\frac{t}{m})+1}\\ =& \frac{m}{t}\{\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{m^n{[n]}_q!}+\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\}\\ \times \sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(0,my)\frac{t^n}{m^n{[n]}_q!}\\ =& \sum _{n=0}^{\mathrm{\infty }}(\frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(x,0)m^{j-k}+{\mathcal{G}}_{k,q}^{(\alpha )}(x,0)\}\\ \times {\mathcal{G}}_{n+1-k,q}(0,my)m^{k-n})\frac{t^n}{{[n]}_q!}.\end{array}$$

Comparing the coefficients of $\frac{t^n}{{[n]}_q!}$, we have (16). The proof of (17) is similar to that of (16). □

3 Explicit relation between the q-Bernoulli polynomials and q-Genocchi polynomials

In this section, we prove two interesting relations between the q-Bernoulli polynomials ${\mathcal{B}}_{n,q}^{(\alpha )}(x,y)$ of order α and the q-Genocchi polynomials ${\mathcal{G}}_{n,q}^{(\alpha )}(x,y)$ of order α.

Theorem 3.1 There is the following relation between q-Genocchi polynomials and q-Bernoulli polynomials

$$\begin{array}{rcl}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)& =& \frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(x,0)m^{j-n}-{\mathcal{G}}_{k,q}^{(\alpha )}(x,0)\}\\ \times {\mathcal{B}}_{n+1-k,q}(0,my)m^{k-n}.\end{array}$$

(18)

Proof From (6), we deduce that

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,y)\frac{t^n}{{[n]}_q!}& =& {\left(\frac{2t}{e_q(t)+1}\right)}^{\alpha }{e}_q(tx)E_q(ty)\\ =& \frac{m}{t}\{\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{m^n{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}(0,my)\frac{t^n}{m^n{[n]}_q!}\\ -\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}(0,my)\frac{t^n}{m^n{[n]}_q!}\}\\ =& \frac{m}{t}\{\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{m^n{[n]}_q!}-\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\}\\ \times \sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}(0,my)\frac{t^n}{m^n{[n]}_q!}\\ =& \sum _{n=0}^{\mathrm{\infty }}(\frac{1}{{[n+1]}_q}\sum _{k=0}^{n+1}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q\{\sum _{j=0}^k{\left[\begin{array}{c}k\\ j\end{array}\right]}_q{\mathcal{G}}_{j,q}^{(\alpha )}(x,0)m^{j-n}-{\mathcal{G}}_{k,q}^{(\alpha )}(x,0)\}\\ \times {\mathcal{B}}_{n+1-k,q}(0,my)m^{k-n})\frac{t^n}{{[n]}_q!}.\end{array}$$

Comparing the coefficients of $\frac{t^n}{{[n]}_q!}$, we have (18). □

Theorem 3.2 There is the following relation between q-Bernoulli polynomials and q-Genocchi polynomials:

$$\begin{array}{rcl}{\mathcal{B}}_{n,q}^{(\alpha )}(x,y)& =& \frac{1}{2}\sum _{r=0}^{n+1}{\left[\begin{array}{c}n+1\\ r\end{array}\right]}_q\frac{1}{{[n+1]}_q}(\sum _{r=0}^k{\left[\begin{array}{c}k\\ r\end{array}\right]}_q{\mathcal{B}}_{k,q}^{(\alpha )}(x,0)m^{k-r}+{\mathcal{B}}_{r,q}^{(\alpha )}(x,0))\\ \times {\mathcal{G}}_{n+1-r,q}(0,my)m^{r-n}.\end{array}$$

(19)

Proof From (2), we obtain

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}^{(\alpha )}(x,y)\frac{t^n}{{[n]}_q!}& =& {\left(\frac{t}{e_q(t)-1}\right)}^{\alpha }{e}_q(tx)E_q(ty)\\ =& \frac{m}{2t}\{{\left(\frac{t}{e_q(t)-1}\right)}^{\alpha }{e}_q(tx)e_q\left(\frac{t}{m}\right)\frac{\frac{2t}{m}}{e_q(\frac{t}{m})+1}{E}_q(\frac{t}{m},my)\\ +{\left(\frac{t}{e_q(t)-1}\right)}^{\alpha }{e}_q(tx)\frac{\frac{2t}{m}}{e_q(\frac{t}{m})+1}{E}_q(\frac{t}{m},my)\}\\ =& \frac{m}{2t}\{\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{m^n{[n]}_q!}+\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,q}^{(\alpha )}(x,0)\frac{t^n}{{[n]}_q!}\}\\ \times \sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n,q}(0,my)\frac{t^n}{m^n{[n]}_q!}\\ =& \frac{m}{2}\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^n{\left[\begin{array}{c}n\\ r\end{array}\right]}_q(\sum _{r=0}^k{\left[\begin{array}{c}k\\ r\end{array}\right]}_q{\mathcal{B}}_{k,q}^{(\alpha )}(x,0)m^{k-r}+{\mathcal{B}}_{r,q}^{(\alpha )}(x,0))\\ \times {\mathcal{G}}_{n-r,q}(0,my)m^{r-n}\frac{1}{{[n]}_q}\frac{t^{n-1}}{{[n-1]}_q!}\\ =& \frac{m}{2}\sum _{n=1}^{\mathrm{\infty }}\{\frac{1}{2}\sum _{r=0}^{n+1}{\left[\begin{array}{c}n+1\\ r\end{array}\right]}_q\frac{1}{{[n+1]}_q}\\ \times (\sum _{r=0}^k{\left[\begin{array}{c}k\\ r\end{array}\right]}_q{\mathcal{B}}_{k,q}^{(\alpha )}(x,0)m^{k-r}+{\mathcal{B}}_{r,q}^{(\alpha )}(x,0))\\ \times {\mathcal{G}}_{n+1-r,q}(0,my)m^{r-n}\}\frac{t^n}{{[n]}_q!}.\end{array}$$

Comparing the coefficients of $\frac{t^n}{{[n]}_q!}$, we have (19). □