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# Some identities on the higher-order-twisted q-Euler numbers and polynomials with weight α

https://doi.org/10.1186/1687-1847-2012-21

• Accepted: 29 February 2012
• Published:

## Abstract

In this article, we introduce some properties of higher-order-twisted q-Euler numbers and polynomials with weight α, and we observe some properties of higher-order-twisted q-Euler numbers and polynomials with weight α for several cases. In particular, by using the the fermionic p-adic q-integral on p , we give a new concept of twisted q-Euler numbers and polynomials with weight α.

2000 Mathematics Subject Classification: 11B68; 11S40; 11S80.

## Keywords

• Euler numbers and polynomials
• q-Euler numbers and polynomials
• higher-order-twisted q-Euler numbers and polynomials with weight α

## 1. Introduction

Let p be a fixed odd prime. Throughout this article p , ${ℚ}_{p}$, , and p , will, respectively, denote the ring of p- adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of ${ℚ}_{p}$. Let be the set of natural numbers and + = {0}. Let ν p be the normalized exponential valuation of p with ${\left|p\right|}_{p}={p}^{-{\nu }_{p}\left(p\right)}={p}^{-1}$ (see [114]). When one speaks of q-extension, q can be regarded as an indeterminate, a complex number q , or p-adic number q p ; it is always clear from context. If q , we assume |q| < 1. If q p , then we assume |1 - q| p < 1 (see [114]).

In this article, we use the notation of q-number as follows (see [114]):
${\left[x\right]}_{q}=\frac{1-{q}^{x}}{1-q}.$

Note that limq→1[x] q = x for any x with |x| p ≤ 1 in the p-adic case.

Let C( p ) be the space of continuous functions on p . For f C( p ), Kim defined the fermionic p-adic q-integral on p as follows (see [6, 7]):
$\begin{array}{ll}\hfill {I}_{-q}\left(f\right)={\int }_{{ℤ}_{p}}f\left(x\right)d{\mu }_{-q}\left(x\right)& =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[{p}^{N}\right]}_{-q}}\sum _{x=0}^{{p}^{N}-1}f\left(x\right){\left(-q\right)}^{x},\phantom{\rule{2em}{0ex}}\\ =\underset{N\to \infty }{\text{lim}}\frac{{\left[2\right]}_{q}}{1+{q}^{{p}^{N}}}\sum _{x=0}^{{p}^{N}-1}f\left(x\right){\left(-q\right)}^{x}.\phantom{\rule{2em}{0ex}}\end{array}$
(1)
From (1), we note that
$q{I}_{-q}\left({f}_{1}\right)+{I}_{-q}\left(f\right)={\left[2\right]}_{q}f\left(0\right),$

where f1(x) = f(x + 1).

It is well known that the ordinary Euler polynomials are defined by
$\frac{2}{{e}^{t}+1}{e}^{xt}={e}^{E\left(x\right)t}=\sum _{n=0}^{\infty }{E}_{n}\left(x\right)\frac{{t}^{n}}{n!},$

with the usual convention of replacing E n (x) by E n (x).

In the special case, x = 0, E n (0) = E n are called the n th Euler numbers (see [114]).

By (2), we get the following recurrence relation as follows:
${E}_{0}=1,\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}{\left(E+1\right)}^{n}+E=\left\{\begin{array}{cc}2,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n=0,\hfill \\ 0,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n>0.\hfill \end{array}\right\$
(2)
Recently, (h, q)-Euler numbers are defined by
${E}_{0,q}^{\left(h\right)}=\frac{2}{1+{q}^{h}},\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}{q}^{h}{\left(q{E}_{q}^{\left(h\right)}+1\right)}^{n}+{E}_{q}^{\left(h\right)}=\left\{\begin{array}{cc}2,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n=0,\hfill \\ 0,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n>0,\hfill \end{array}\right\$

with the usual convention about replacing ${\left({E}_{q}^{\left(h\right)}\right)}^{n}$ by ${E}_{n,q}^{\left(h\right)}$ (see [116]).

Note that ${\mathrm{lim}}_{q\to 1}{E}_{n,q}^{\left(h\right)}={E}_{n}$.

Let ${T}_{p}={\cup }_{N\ge 1}{C}_{{p}^{N}}={\mathrm{lim}}_{N\to \infty }{C}_{{p}^{N}}$, where ${C}_{{p}^{N}}=\left\{w|{w}^{{p}^{N}}=1\right\}$ is the cyclic group of order p N . For w T p , we denote by ϕ w : p p the locally constant function x w x .

For α and w T p , the twisted q-Euler numbers with weight α are also defined by
${Ẽ}_{0,q,w}^{\left(\alpha \right)}=\frac{{\left[2\right]}_{q}}{wq+1},\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}wq{\left({q}^{\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}+1\right)}^{n}+{Ẽ}_{n,q,w}^{\left(\alpha \right)}=\left\{\begin{array}{cc}{\left[2\right]}_{q},\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n=0,\hfill \\ 0,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n>0,\hfill \end{array}\right\$

with the usual convention about replacing ${\left({Ẽ}_{q,w}^{\left(\alpha \right)}\right)}^{n}$ by ${Ẽ}_{n,q,w}^{\left(\alpha \right)}$ (see [2, 5]).

The main purpose of this article is to present a systemic study of some families of higher-order-twisted q-Euler numbers and polynomials with weight α. In Section 2, we investigate higher-order-twisted q-Euler numbers and polynomials with weight α and establish interesting properties. In Sections 3, 4, and 5, we observe some properties for special cases.

## 2. Higher-order-twisted q-Euler numbers and polynomials with weight α

For h , α, k , w T p and n +, let us consider the expansion of higher-order-twisted q-Euler polynomials with weight α as follows:
$\begin{array}{c}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)\\ =\underset{k-\text{times}}{\underset{⏟}{{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}w}}{\sum }_{i=1}^{k}{x}_{i}{\left[\sum _{i=1}^{k}{x}_{i}+x\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-1\right)+\cdots +{x}_{k}\left(h-k\right)}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right).\end{array}$
(3)
From (1) and (3), we note that
${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)=\frac{{\left[2\right]}_{q}^{k}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}\begin{array}{c}n\hfill \\ l\hfill \end{array}\hfill \end{array}\right){\left(-1\right)}^{l}\frac{{q}^{\alpha lx}}{\left(1+w{q}^{\alpha l+h}\right)\cdots \left(1+w{q}^{\alpha l+h-k+1}\right)}.$
(4)

In the special case, x = 0 ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|0\right)={Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k\right)$ are called the higher-order-twisted q-Euler numbers with weight α.

By (3), we get
${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k\right)=\left({q}^{\alpha }-1\right){Ẽ}_{n+1,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k\right).$
(5)

From (5) and mathematical induction, we get the following theorem.

Theorem 1. For α, k and n +, we have
$\begin{array}{c}\sum _{i=0}^{n-1}{\left(-1\right)}^{i-1}{\left({q}^{\alpha }-1\right)}^{n-1-i}{Ẽ}_{n-i,q,w}^{\left(\alpha \right)}\left(h,k\right)\\ ={\left({q}^{\alpha }-1\right)}^{n-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k\right)+{\left(-1\right)}^{n}{E}_{1,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k\right).\end{array}$
For complex number q p , m +, we get the following;
$\begin{array}{ll}\hfill {q}^{\alpha \left({x}_{1}+\cdots +{x}_{k+1}\right)m}& ={\left(1-\left(1-{q}^{\alpha \left({x}_{1}+\cdots +{x}_{k+1}\right)}\right)\right)}^{m}\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{m}{\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}\left(1-{q}^{\alpha \left({x}_{1}+\cdots +{x}_{k+1}\right)}\right)}^{l}\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{m}\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{\left(1-{q}^{\alpha }\right)}^{l}\frac{{\left(1-{q}^{\alpha \left({x}_{1}+\cdots +{x}_{k+1}\right)}\right)}^{l}}{{\left(1-{q}^{\alpha }\right)}^{l}}\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{m}\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{\left(1-{q}^{\alpha }\right)}^{l}{\left[{x}_{1}+{x}_{2}+\cdots +{x}_{k+1}\right]}_{{q}^{\alpha }}^{{}_{l}}.\phantom{\rule{2em}{0ex}}\end{array}$
From (3), (4), and above property, we have
$\begin{array}{l}{Ẽ}_{0,q,w}^{\left(\alpha \right)}\left(m\alpha ,k+1\right)\phantom{\rule{2em}{0ex}}\\ ={\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{\sum }_{j=1}^{k+1}{x}_{j}}{q}^{{\sum }_{j=1}^{k+1}\left(m\alpha -j\right){x}_{j}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k+1}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{m}\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{l}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{\sum }_{j=1}^{k+1}{x}_{j}}{\left[\sum _{j=1}^{k+1}{x}_{j}\right]}_{{q}^{\alpha }}^{l}{q}^{-{\sum }_{j=1}^{k+1}j{x}_{j}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k+1}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{m}\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{l}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(0,k+1\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[2\right]}_{q}^{k+1}}{\left(1+w{q}^{\alpha m}\right)\left(1+w{q}^{\alpha m-1}\right)\dots \left(1+w{q}^{\alpha m-k}\right)}.\phantom{\rule{2em}{0ex}}\end{array}$
(6)
From (3), we can derive the following equation.
$\begin{array}{c}\sum _{j=0}^{i}\left(\begin{array}{c}i\hfill \\ j\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{j}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{\sum }_{s=1}^{k}{x}_{s}}{\left[\sum _{s=1}^{k}{x}_{s}\right]}_{{q}^{\alpha }}^{n-i+j}{q}^{{\sum }_{s=1}^{k}\left(h-\alpha -s\right){x}_{s}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ ={\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{\sum }_{s=1}^{k}{x}_{s}}{\left[\sum _{s=1}^{k}{x}_{s}\right]}_{{q}^{\alpha }}^{n-i}{q}^{{\sum }_{s=1}^{k}\left(h-s\right){x}_{s}}{q}^{\alpha \left({\sum }_{s=1}^{k}{x}_{s}\right)\left(i-1\right)}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ =\sum _{j=0}^{i-1}{\left({q}^{\alpha }-1\right)}^{j}\left(\begin{array}{c}i-1\hfill \\ j\hfill \end{array}\right){Ẽ}_{n-i+j,q,w}^{\left(\alpha \right)}\left(h,k\right).\end{array}$
(7)
By (3), (4), (5), and (6), we see that
$\sum _{j=0}^{i}{\left({q}^{\alpha }-1\right)}^{j}\left(\begin{array}{c}i\hfill \\ j\hfill \end{array}\right){Ẽ}_{n-1+j,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k\right)=\sum _{j=0}^{i-1}{\left({q}^{\alpha }-1\right)}^{j}\left(\begin{array}{c}i-1\hfill \\ j\hfill \end{array}\right){Ẽ}_{n-i+j,q,w}^{\left(\alpha \right)}\left(h,k\right).$

Therefore, we obtain the following theorem.

Theorem 2. For α, k and n, i +, we have
$\sum _{j=0}^{i}\left(\begin{array}{c}i\hfill \\ j\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{j}{Ẽ}_{n-i+j,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k\right)=\sum _{j=0}^{i-1}{\left({q}^{\alpha }-1\right)}^{j}\left(\begin{array}{c}i-1\hfill \\ j\hfill \end{array}\right){Ẽ}_{n-i+j,q,w}^{\left(\alpha \right)}\left(h,k\right).$
By simple calculation, we easily see that
$\sum _{j=0}^{m}\left(\begin{array}{c}m\hfill \\ j\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{j}{Ẽ}_{j,q,w}^{\left(\alpha \right)}\left(0,k\right)=\frac{{\left[2\right]}_{q}^{k}}{\left(1+w{q}^{\alpha m}\right)\left(1+w{q}^{\alpha m-1}\right)\cdots \left(1+w{q}^{\alpha m-k+1}\right)}.$

## 3. Polynomials ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)$

We now consider the polynomials ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)$ (in q x ) by
$\begin{array}{c}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)\\ =\underset{k-\text{times}}{\underset{⏟}{{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}}}{w}^{{x}_{1}+\cdots +{x}_{k}}{\left[x+\sum _{i=1}^{k}{x}_{i}\right]}_{{q}^{\alpha }}^{n}{q}^{-{\sum }_{j=1}^{k}j{x}_{j}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right).\end{array}$
(8)
By (8) and (4), we get
${\left({q}^{\alpha }-1\right)}^{n}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)={\left[2\right]}_{q}^{k}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){q}^{\alpha lx}{\left(-1\right)}^{n-1}\frac{1}{\left(1+w{q}^{\alpha l}\right)\cdots \left(1+w{q}^{\alpha l-k+1}\right)}.$
(9)
From (8) and (9), we can derive the following equation.
$\begin{array}{c}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{q}^{{\sum }_{j=1}^{k}\left(\alpha n-j\right){x}_{j}+\alpha nx}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ =\sum _{j=0}^{n}\left(\begin{array}{c}n\hfill \\ j\hfill \end{array}\right){\left[\alpha \right]}_{q}^{j}{\left(q-1\right)}^{j}{Ẽ}_{j,q,w}^{\left(\alpha \right)}\left(0,k|x\right),\end{array}$
and
$\begin{array}{l}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{q}^{{\sum }_{j=1}^{k}\left(\alpha n-j\right){x}_{j}+\alpha nx}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[2\right]}_{q}^{k}{q}^{\alpha nx}}{\left(1+w{q}^{\alpha n}\right)\cdots \left(1+w{q}^{\alpha n-k+1}\right)}.\phantom{\rule{2em}{0ex}}\end{array}$
(10)

Therefore, by (9) and (10), we obtain the following theorem.

Theorem 3. For α and n, k +, we have
${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)=\frac{{\left[2\right]}_{q}^{k}}{{\left[\alpha \right]}_{q}^{n}{\left(1-q\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}\frac{1}{{\left(-w{q}^{\alpha l-k+1}:q\right)}_{k}},$
and
$\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[\alpha \right]}_{q}^{l}{\left(q-1\right)}^{l}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(0,k|x\right)=\frac{{q}^{\alpha nx}{\left[2\right]}_{q}^{k}}{{\left(-w{q}^{\alpha n-k+1}:q\right)}_{k}},$

where (a : q)0 = 1 and (a : q) k = (1 - a)(1 - aq) (1 - aqk-1).

Let d with d ≡ 1 (mod 2). Then we have
$\begin{array}{l}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{\left[x+\sum _{j=1}^{k}{x}_{j}\right]}_{{q}^{\alpha }}^{n}{q}^{-{\sum }_{j=1}^{k}j{x}_{j}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}^{k}}\sum _{{a}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{a}_{k}=0}^{d-1}{w}^{{a}_{1}+\cdots +{a}_{k}}{q}^{-{\sum }_{j=2}^{k}\left(j-1\right){a}_{j}}{\left(-1\right)}^{{\sum }_{j=1}^{k}{a}_{j}}×\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{d\left({x}_{1}+\cdots +{x}_{k}\right)}{\left[\frac{x+{\sum }_{j=1}^{k}{a}_{j}}{d}+\sum _{j=1}^{k}{x}_{j}\right]}_{{q}^{\alpha d}}^{n}{q}^{-d{\sum }_{j=1}^{k}j{x}_{j}}d{\mu }_{-{q}^{d}}\left({x}_{1}\right)\cdots d{\mu }_{-{q}^{d}}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\end{array}$
(11)

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

Theorem 4. For d with d ≡ 1 (mod 2), we have
$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}^{k}}\sum _{{a}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{a}_{k}=0}^{d-1}{\left(-w\right)}^{{a}_{1}+\cdots +{a}_{k}}{q}^{-{\sum }_{j=2}^{k}\left(j-1\right){a}_{j}}{Ẽ}_{n,{q}^{d},{w}^{d}}^{\left(\alpha \right)}\left(0,k|\frac{x+{a}_{1}+\cdots +{a}_{k}}{d}\right).\phantom{\rule{2em}{0ex}}\end{array}$
Moreover,
$\begin{array}{c}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|dx\right)\\ =\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}^{k}}\sum _{{a}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{a}_{k}=0}^{d-1}{\left(-w\right)}^{{a}_{1}+\cdots +{a}_{k}}{q}^{-{\sum }_{j=2}^{k}\left(j-1\right){a}_{j}}{Ẽ}_{n,{q}^{d},{w}^{d}}^{\left(\alpha \right)}\left(0,k|x+\frac{{a}_{1}+\cdots +{a}_{k}}{d}\right).\end{array}$
By (8), we get
$\begin{array}{l}{\stackrel{˜}{E}}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x=\sum _{l=0}^{n}\left(\begin{array}{c}n\\ l\end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{n-l}{q}^{\alpha lx}{\stackrel{˜}{E}}_{l,q,w}^{\left(\alpha \right)}\left(0,k\right)\\ \phantom{\rule{1em}{0ex}}={\left({\left[x\right]}_{{q}^{\alpha }}+{q}^{\alpha x}{\stackrel{˜}{E}}_{q,w}^{\left(\alpha \right)}\left(0,k\right)\right)}^{n},\end{array}$

where ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|0\right)={Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k\right)$.

Thus, we note that
$\begin{array}{l}{\stackrel{˜}{E}}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x+y\right)=\sum _{l=0}^{n}\left(\begin{array}{c}n\\ l\end{array}\right){\left[y\right]}_{{q}^{\alpha }}^{n-l}{q}^{\alpha ly}{\stackrel{˜}{E}}_{l,q,w}^{\left(\alpha \right)}\left(0,k|x\right)\\ \phantom{\rule{1em}{0ex}}={\left({\left[y\right]}_{{q}^{\alpha }}+{q}^{\alpha y}{\stackrel{˜}{E}}_{q,w}^{\left(\alpha \right)}\left(0,k|x\right)\right)}^{n}.\end{array}$

## 4. Polynomials ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)$

Let us define polynomials ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)$ as follows:
${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)={\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right).$
(12)
From (12), we have
${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)=\frac{{\left[2\right]}_{q}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}\frac{1}{\left(1+w{q}^{\alpha l+h}\right)}.$
By the calculation of the fermionic p-adic q- integral on p , we see that
$\begin{array}{c}{q}^{\alpha x}{\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right)\\ =\left({q}^{\alpha }-1\right){\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n+1}{q}^{{x}_{1}\left(h-\alpha -1\right)}d{\mu }_{-q}\left({x}_{1}\right)+{\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-\alpha -1\right)}d{\mu }_{-q}\left({x}_{1}\right).\end{array}$
(13)

Thus, by (13), we obtain the following theorem.

Theorem 5. For α and h , we have
${q}^{\alpha x}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)=\left({q}^{\alpha }-1\right){Ẽ}_{n+1,q,w}^{\left(\alpha \right)}\left(h-\alpha ,1|x\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h-\alpha ,1|x\right).$
It is easy to show that
$\begin{array}{ll}\hfill {Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)& ={\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{n-1}{q}^{\alpha lx}{\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[{x}_{1}\right]}_{{q}^{\alpha }}^{l}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{n-1}{q}^{\alpha lx}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(h,1\right)\phantom{\rule{2em}{0ex}}\\ ={\left({q}^{\alpha x}{Ẽ}_{q,w}^{\left(\alpha \right)}\left(h,1\right)+{\left[x\right]}_{{q}^{\alpha }}\right)}^{n},\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{2.77695pt}{0ex}}n\ge 1,\phantom{\rule{2em}{0ex}}\end{array}$
(14)

with the usual convention about replacing ${\left({Ẽ}_{q,w}^{\left(\alpha \right)}\left(h,1\right)\right)}^{n}$ by ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)$.

From qI-q(f1) + I-q(f) = [2] q f(0), we have
$\begin{array}{l}w{q}^{h}{\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}+1\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right)+{\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right)\phantom{\rule{2em}{0ex}}\\ ={\left[2\right]}_{q}{\left[x\right]}_{{q}^{\alpha }}^{n}.\phantom{\rule{2em}{0ex}}\end{array}$
(15)
By (13) and (15), we get
$w{q}^{h}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x+1\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)={\left[2\right]}_{q}{\left[x\right]}_{{q}^{\alpha }}^{n}.$
(16)
For x = 0 in (16), we have
$w{q}^{h}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|1\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)=\left\{\begin{array}{cc}{\left[2\right]}_{q},\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n=0,\hfill \\ 0,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n>0.\hfill \end{array}\right\$
(17)

Therefore, by (14) and (17), we obtain the following theorem.

Theorem 6. For h and n +, we have
$w{q}^{h}{\left({q}^{\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}\left(h,1\right)+1\right)}^{n}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)=\left\{\begin{array}{cc}{\left[2\right]}_{q},\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n=0,\hfill \\ 0,\hfill & \text{if}\phantom{\rule{2.77695pt}{0ex}}n>0,\hfill \end{array}\right\$

with the usual convention about replacing ${\left({Ẽ}_{q,w}^{\left(\alpha \right)}\left(h,1\right)\right)}^{n}$ by ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)$.

From the fermionic p-adic q-integral on p , we easily get
${Ẽ}_{0,q,w}^{\left(\alpha \right)}\left(h,1\right)={\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{q}^{{x}_{1}\left(h-1\right)}d{\mu }_{-q}\left({x}_{1}\right)=\frac{{\left[2\right]}_{q}}{{\left[2\right]}_{w{q}^{h}}}.$
By (12), we see that
$\begin{array}{l}{Ẽ}_{n,{q}^{-1},{w}^{-1}}^{\left(\alpha \right)}\left(h,1|1-x\right)={\int }_{{ℤ}_{p}}{w}^{-1}{\left[1-x+{x}_{1}\right]}_{{q}^{-\alpha }}^{n}{q}^{-{x}_{1}\left(h-1\right)}d{\mu }_{-{q}^{-1}}\left({x}_{1}\right)\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}\frac{{\left[2\right]}_{q}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}\frac{1}{1+w{q}^{\alpha l+h}}\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)\phantom{\rule{2em}{0ex}}\end{array}$
(18)

Therefore, by (18), we obtain the following theorem.

Theorem 7. For α , h and n +, we have
${Ẽ}_{n,{q}^{-1},{w}^{-1}}^{\left(\alpha \right)}\left(h,1|1-x\right)={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right).$
In particular, for x = 1, we get
$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|1\right)\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n+1}{q}^{\alpha n-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}n\ge 1.\phantom{\rule{2em}{0ex}}\end{array}$
Let d with d ≡ 1 (mod 2). Then we have
$\begin{array}{l}{\int }_{{ℤ}_{p}}{w}^{{x}_{1}}{q}^{{x}_{1}\left(h-1\right)}{\left[x+{x}_{1}\right]}_{{q}^{\alpha }}^{n}d{\mu }_{-q}\left({x}_{1}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}}\sum _{a=0}^{d-1}{w}^{a}{q}^{ha}{\left(-1\right)}^{a}{\int }_{{ℤ}_{p}}{w}^{d{x}_{1}}{\left[\frac{x+a}{d}+{x}_{1}\right]}_{{q}^{\alpha d}}^{n}{q}^{{x}_{1}\left(h-1\right)d}d{\mu }_{-{q}^{d}}\left({x}_{1}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(19)

Therefore, by (19), we obtain the following theorem.

Theorem 8 (Multiplication formula). For d with d ≡ 1 (mod 2), we have
${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)=\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}}\sum _{a=0}^{d-1}{w}^{a}{q}^{ha}{\left(-1\right)}^{q}{Ẽ}_{n,{q}^{d},{w}^{d}}^{\left(\alpha \right)}\left(h,1|\frac{x+a}{d}\right).$

## 5. Polynomials ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)$and k= h

In (3), we know that
$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)\phantom{\rule{2em}{0ex}}\\ ={\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{\left[{x}_{1}+\cdots +{x}_{k}+x\right]}_{{q}^{\alpha }}^{n}{q}^{\left(h-1\right){x}_{1}+\cdots +\left(h-k\right){x}_{k}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right).\phantom{\rule{2em}{0ex}}\end{array}$
Thus, we get
${\left({q}^{\alpha }-1\right)}^{n}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)={\left[2\right]}_{q}^{k}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{n-l}\frac{{q}^{\alpha lx}}{\left(1+w{q}^{\alpha l+h}\right)\cdots \left(1+w{q}^{\alpha l+h-k+1}\right)},$
and
$\begin{array}{l}w{q}^{h}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{\left[x+1+\sum _{i=1}^{k}{x}_{i}\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{i=1}^{k}\left(h-i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =-{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{\left[x+\sum _{i=1}^{k}{x}_{i}\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{i=1}^{k}\left(h-i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+{\left[2\right]}_{q}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{2}+\cdots +{x}_{k}}{\left[x+\sum _{i=2}^{k}{x}_{i}\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{i=2}^{k}\left(h-i\right){x}_{i}}d{\mu }_{-q}\left({x}_{2}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(20)

Therefore, by (3) and (20), we obtain the following theorem.

Theorem 9. For h , α and n +, we have
$w{q}^{h}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x+1\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)={\left[2\right]}_{q}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h-1,k-1|x\right).$
Note that
$\begin{array}{l}{q}^{\alpha x}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{w}_{k}}\left[x+\sum _{i=1}^{k}{x}_{i}{\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{i=1}^{k}\left(h-i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ =\left({q}^{\alpha }-1\right){\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}{\left[x+\sum _{i=1}^{k}{x}_{i}\right]}_{{q}^{\alpha }}^{n+1}{q}^{{\sum }_{i=1}^{k}\left(h-\alpha -i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ \phantom{\rule{1em}{0ex}}+{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}\left[x+\sum _{i=1}^{k}{x}_{i}{\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{i=1}^{k}\left(h-\alpha -i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ =\left({q}^{\alpha }-1\right){\stackrel{˜}{E}}_{n+1,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k|x\right)+{\stackrel{˜}{E}}_{n,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k|x\right).\end{array}$
(21)

Therefore, by (21), we obtain the following theorem.

Theorem 10. For n +, we have
${q}^{\alpha x}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)=\left({q}^{\alpha }-1\right){Ẽ}_{n+1,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k|x\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h-\alpha ,k|x\right).$
Let d with d ≡ 1 (mod 2). Then we get
$\begin{array}{l}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{x}_{1}+\cdots +{x}_{k}}\left[x+\sum _{j=1}^{k}{x}_{j}{\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{j=1}^{k}\left(h-j\right){x}_{j}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\\ =\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}^{k}}\sum _{{a}_{1},\cdots ,{a}_{k}=0}^{d-1}{w}^{{a}_{1}+\cdots +{a}_{k}}{q}^{h{\sum }_{j=1}^{k}aj-{\sum }_{j=2}^{k}\left(j-1\right){a}_{j}}{\left(-1\right)}^{{\sum }_{j=1}^{k}{a}_{j}}×\\ \phantom{\rule{1em}{0ex}}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{d\left({x}_{1}+\cdots +{x}_{k}\right)}{\left[\frac{x+{\sum }_{j=1}^{k}{a}_{j}}{d}+\sum _{j=1}^{k}{x}_{j}\right]}_{{q}^{\alpha d}}^{n}{q}^{d{\sum }_{j=1}^{k}\left(h-j\right){x}_{j}}d{\mu }_{-{q}^{d}}\left({x}_{1}\right)\cdots d{\mu }_{-{q}^{d}}\left({x}_{k}\right).\end{array}$
(22)

Therefore, by (22), we obtain the following theorem.

Theorem 11. For d with d ≡ 1 (mod 2), we have
$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|dx\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[d\right]}_{{q}^{\alpha }}^{n}}{{\left[d\right]}_{-q}^{k}}\sum _{{a}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{a}_{k}=0}^{d-1}{w}^{{a}_{1}+\cdots +{a}_{k}}{q}^{h{\sum }_{j=1}^{k}aj-{\sum }_{j=2}^{k}\left(j-1\right){a}_{j}}{\left(-1\right)}^{{\sum }_{j=1}^{k}{a}_{j}}{Ẽ}_{n,{q}^{d},{w}^{d}}^{\left(\alpha \right)}\left(h,k|x+\frac{{a}_{1}+\cdots +{a}_{k}}{d}\right).\phantom{\rule{2em}{0ex}}\end{array}$
Let ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k,k|x\right)={Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right)$. Then we get
${\left({q}^{\alpha }-1\right)}^{n}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right)=\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{n-l}{q}^{\alpha lx}\frac{{\left[2\right]}_{q}^{k}}{\left(1+w{q}^{\alpha l+k}\right)\cdots \left(1+w{q}^{\alpha l+1}\right)},$
and
$\begin{array}{l}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{-\left({x}_{1}+\cdots +{x}_{k}\right)}{\left[k-x+\sum _{i=1}^{k}{x}_{i}\right]}_{{q}^{-\alpha }}^{n}{q}^{-{\sum }_{i=1}^{k}\left(k-i\right){x}_{i}}d{\mu }_{-{q}^{-1}}\left({x}_{1}\right)\cdots d{\mu }_{-{q}^{-1}}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{q}^{\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}}{{\left(1-{q}^{-\alpha }\right)}^{n}}{\left[2\right]}_{q}^{k}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}\frac{1}{\left(1+w{q}^{\alpha l+1}\right)\cdots \left(1+w{q}^{\alpha l+k}\right)}\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n}{q}^{n\alpha }{q}^{\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}\frac{{\left[2\right]}_{q}^{k}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\frac{\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}}{\left(1+w{q}^{\alpha l+1}\right)\cdots \left(1+w{q}^{\alpha l+k}\right)}\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n}{q}^{\alpha n+\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right).\phantom{\rule{2em}{0ex}}\end{array}$
(23)

Therefore, by (23), we obtain the following theorem.

Theorem 12. For n +, we have
${Ẽ}_{n,{q}^{-1},{w}^{-1}}^{\left(\alpha \right)}\left(k|k-x\right)={\left(-1\right)}^{n}{w}^{k}{q}^{\alpha n+\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right).$
Let x = k in Theorem 12. Then we see that
${Ẽ}_{n,{q}^{-1},{w}^{-1}}^{\left(\alpha \right)}\left(k|0\right)={\left(-1\right)}^{n}{w}^{k}{q}^{\alpha n+\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|k\right).$
(24)
From (15), we note that
$w{q}^{k}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x+1\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right)={\left[2\right]}_{q}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k-1|x\right).$
(25)
It is easy to show that
${\left({q}^{\alpha }-1\right)}^{n}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|0\right)=\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^{l+n}\frac{{\left[2\right]}_{q}^{k}}{\left(1+w{q}^{\alpha l+1}\right)\cdots \left(1+w{q}^{\alpha l+k}\right)}.$
By simple calculation, we get
$\begin{array}{l}\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{l}{\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{\sum }_{i=1}^{k}{x}_{k}}{\left[\sum _{i=1}^{k}{x}_{k}\right]}_{{q}^{\alpha }}^{l}{q}^{{\sum }_{l=i}^{k}\left(k-i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[2\right]}_{q}^{k}}{\left(1+w{q}^{\alpha n+k}\right)\left(1+w{q}^{\alpha n+k-1}\right)\cdots \left(1+w{q}^{\alpha n+1}\right)}.\phantom{\rule{2em}{0ex}}\end{array}$
(26)
From (26), we note that
$\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left({q}^{\alpha }-1\right)}^{l}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(k|0\right)=\frac{{\left[2\right]}_{q}^{k}}{\left(1+w{q}^{\alpha n+k}\right)\left(1+w{q}^{\alpha n+k-1}\right)\cdots \left(1+w{q}^{\alpha n+1}\right)},$
and
$\begin{array}{ll}\hfill {Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right)& ={\int }_{{ℤ}_{p}}\cdots {\int }_{{ℤ}_{p}}{w}^{{\sum }_{i=1}^{k}{x}_{k}}{\left[x+\sum _{i=1}^{k}{x}_{k}\right]}_{{q}^{\alpha }}^{n}{q}^{{\sum }_{i=1}^{k}\left(k-i\right){x}_{i}}d{\mu }_{-q}\left({x}_{1}\right)\cdots d{\mu }_{-q}\left({x}_{k}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{n}\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{n-l}{q}^{\alpha lx}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(k|0\right)\phantom{\rule{2em}{0ex}}\\ ={\left({q}^{x\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}\left(k|0\right)+{\left[x\right]}_{{q}^{\alpha }}\right)}^{n},n\in {ℤ}_{+},\phantom{\rule{2em}{0ex}}\end{array}$

with the usual convention about replacing ${\left({Ẽ}_{q,w}^{\left(\alpha \right)}\left(k|0\right)\right)}^{n}$ by ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|0\right)$.

Put x = 0 in (25), we get
$w{q}^{k}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|1\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|0\right)={\left[2\right]}_{q}{Ẽ}_{n,q}^{\left(\alpha \right)}\left(k-1|0\right).$
Thus, we have
$w{q}^{k}{\left({q}^{\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}\left(k|0\right)+1\right)}^{n}+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|0\right)={\left[2\right]}_{q}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k-1|0\right),$

with the usual convention about replacing ${\left({Ẽ}_{q,w}^{\left(\alpha \right)}\left(k|0\right)\right)}^{n}$ by ${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|0\right)$.

## Declarations

### Acknowledgements

The authors express their gratitude to the referee for his/her valuable comments.

## Authors’ Affiliations

(1)
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea

## References

1. Ryoo CS: On the generalized Barnes type multiple q -Euler polynomials twisted by ramified roots of unity. Proc Jangjeon Math Soc 2010, 13: 255–263.
2. Ryoo CS: A note on the weighted q -Euler numbers and polynomials. Adv Stud Contemp Math 2011, 21: 47–54.
3. Moon E-J, Rim S-H, Jin J-H, Lee S-J: On the symmetric properties of higher-order twisted q -Euler numbers and polynomials. Adv Diff Equ 2010, 2010: 8. (Art ID 765259)
4. Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv Stud Contemp Math 2009, 19: 39–57.
5. Jang LC: A note on Nörlund-type twisted q -Euler polynomials and numbers of higher order associated with fermionic invariant q -integrals. J Inequal Appl 2010, 2010: 12. (Art ID 417452)
6. Kim T: q-Volkenborn integration. Russ J Math Phys 2002, 9: 288–299.
7. Kim T: A note on q-Volkenborn integration. Proc Jangjeon Math Soc 2005, 8: 13–17.
8. Kim T, Choi J, Kim YH: On extended Carlitz's type q -Euler numbers and polynomials. Adv Stud Contemp Math 2010, 20: 499–505.
9. Kim T: The modified q -Euler numbers and polynomials. Adv Stud Contemp Math 2008, 16: 161–170.
10. Kim T: q -Euler numbers and polynomials associated with p -adic q -integrals. J Nonlinear Math Phys 2007, 14: 15–27. 10.2991/jnmp.2007.14.1.3
11. Kim T: Note on the Euler q -zeta functions. J Number Theory 2009, 129: 1798–1804. 10.1016/j.jnt.2008.10.007
12. Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on p . Russ J Math Phys 2009, 16: 484–491. 10.1134/S1061920809040037
13. Kim T: Barnes type multiple q -zeta function and q -Euler polynomials. J Phys A: Math Theor 2010, 43: 11. (Art ID 255201)Google Scholar
14. Kim T, Choi J, Kim YH, Ryoo CS: On the fermionic p -adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials. J Inequal Appl 2010, 2010: 12. (Art ID 864247)Google Scholar
15. Kim T, Choi J, Kim YH, Ryoo CS: A note on the weighted p -adic q -Euler measure on p . Adv Stud Contemp Math 2011, 21: 35–40.
16. Kurt V: A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Appl Math Sci 2009, 3: 53–56.