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

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 ℤ _p , we give a new concept of twisted q-Euler numbers and polynomials with weight α.

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

1. Introduction

Let p be a fixed odd prime. Throughout this article ℤ _p , ${\mathbb{Q}}_p$, ℂ, and ℂ _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 ${\mathbb{Q}}_p$. Let ℕ be the set of natural numbers and ℤ₊ = ℕ∪{0}. Let ν _p be the normalized exponential valuation of ℂ _p with ${\left|p\right|}_p=p^{-{\nu }_p\left(p\right)}=p^{-1}$ (see [1–14]). When one speaks of q-extension, q can be regarded as an indeterminate, a complex number q ∈ ℂ, or p-adic number q ∈ ℂ _p ; it is always clear from context. If q ∈ ℂ, we assume |q| < 1. If q ∈ ℂ _p , then we assume |1 - q| _p < 1 (see [1–14]).

In this article, we use the notation of q-number as follows (see [1–14]):

$${\left[x\right]}_q=\frac{1-q^x}{1-q}.$$

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

Let C(ℤ _p ) be the space of continuous functions on ℤ _p . For f ∈ C(ℤ _p ), Kim defined the fermionic p-adic q-integral on ℤ _p as follows (see [6, 7]):

$$\begin{array}{ll}\hfill I_{-q}\left(f\right)={\int }_{{\mathbb{Z}}_p}f\left(x\right)d{\mu }_{-q}\left(x\right)& =\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[p^N\right]}_{-q}}\sum _{x=0}^{p^N-1}f\left(x\right){\left(-q\right)}^x,\\ =\underset{N\to \infty }{\text{lim}}\frac{{\left[2\right]}_q}{1+q^{p^N}}\sum _{x=0}^{p^N-1}f\left(x\right){\left(-q\right)}^x.\end{array}$$

(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 Eⁿ(x) by E _n (x).

In the special case, x = 0, E _n (0) = E _n are called the n th Euler numbers (see [1–14]).

By (2), we get the following recurrence relation as follows:

$$E_0=1,\text{and}{\left(E+1\right)}^n+E=\left\{\begin{array}{cc}2,\hfill & \text{if}n=0,\hfill \\ 0,\hfill & \text{if}n>0.\hfill \end{array}\right.$$

(2)

Recently, (h, q)-Euler numbers are defined by

$$E_{0,q}^{\left(h\right)}=\frac{2}{1+q^h},\text{and}{q}^h{\left(q{E}_q^{\left(h\right)}+1\right)}^n+E_q^{\left(h\right)}=\left\{\begin{array}{cc}2,\hfill & \text{if}n=0,\hfill \\ 0,\hfill & \text{if}n>0,\hfill \end{array}\right.$$

with the usual convention about replacing ${\left(E_q^{\left(h\right)}\right)}^n$ by $E_{n,q}^{\left(h\right)}$ (see [1–16]).

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

Let $T_p={\cup }_{N\ge 1}{C}_{p^N}={\lim }_{N\to \infty }{C}_{p^N}$, 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 : ℤ _p → ℂ _p the locally constant function x ↦ w^x.

For α ∈ ℕ and w ∈ T _p , the twisted q-Euler numbers with weight α are also defined by

$${Ẽ}_{0,q,w}^{\left(\alpha \right)}=\frac{{\left[2\right]}_q}{wq+1},\text{and}wq{\left(q^{\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}+1\right)}^n+{Ẽ}_{n,q,w}^{\left(\alpha \right)}=\left\{\begin{array}{cc}{\left[2\right]}_q,\hfill & \text{if}n=0,\hfill \\ 0,\hfill & \text{if}n>0,\hfill \end{array}\right.$$

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 ∈ ℤ, α, k ∈ ℕ, w ∈ T _p and n ∈ ℤ₊, let us consider the expansion of higher-order-twisted q-Euler polynomials with weight α as follows:

$$\begin{array}{c}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)\\ =\underset{k-\text{times}}{\underbrace{{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}w}}{\sum }_{i=1}^k{x}_i{\left[\sum _{i=1}^k{x}_i+x\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-1\right)+\cdots +x_k\left(h-k\right)}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right).\end{array}$$

(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 ∈ ℕ and n ∈ ℤ₊, 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 ∈ ℂ _p , m ∈ ℤ₊, we get the following;

$$\begin{array}{ll}\hfill q^{\alpha \left(x_1+\cdots +x_{k+1}\right)m}& ={\left(1-\left(1-q^{\alpha \left(x_1+\cdots +x_{k+1}\right)}\right)\right)}^m\\ =\sum _{l=0}^m{\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^l\left(1-q^{\alpha \left(x_1+\cdots +x_{k+1}\right)}\right)}^l\\ =\sum _{l=0}^m\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^l{\left(1-q^{\alpha }\right)}^l\frac{{\left(1-q^{\alpha \left(x_1+\cdots +x_{k+1}\right)}\right)}^l}{{\left(1-q^{\alpha }\right)}^l}\\ =\sum _{l=0}^m\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^l{\left(1-q^{\alpha }\right)}^l{\left[x_1+x_2+\cdots +x_{k+1}\right]}_{q^{\alpha }}^{{}_l}.\end{array}$$

From (3), (4), and above property, we have

$$\begin{array}{l}{Ẽ}_{0,q,w}^{\left(\alpha \right)}\left(m\alpha ,k+1\right)\\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{{\sum }_{j=1}^{k+1}{x}_j}{q}^{{\sum }_{j=1}^{k+1}\left(m\alpha -j\right)x_j}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_{k+1}\right)\\ =\sum _{l=0}^m\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(q^{\alpha }-1\right)}^l{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{{\sum }_{j=1}^{k+1}{x}_j}{\left[\sum _{j=1}^{k+1}{x}_j\right]}_{q^{\alpha }}^l{q}^{-{\sum }_{j=1}^{k+1}j{x}_j}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_{k+1}\right)\\ =\sum _{l=0}^m\left(\begin{array}{c}m\hfill \\ l\hfill \end{array}\right){\left(q^{\alpha }-1\right)}^l{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(0,k+1\right)\\ =\frac{{\left[2\right]}_q^{k+1}}{\left(1+w{q}^{\alpha m}\right)\left(1+w{q}^{\alpha m-1}\right)\dots \left(1+w{q}^{\alpha m-k}\right)}.\end{array}$$

(6)

From (3), we can derive the following equation.

$$\begin{array}{c}\sum _{j=0}^i\left(\begin{array}{c}i\hfill \\ j\hfill \end{array}\right){\left(q^{\alpha }-1\right)}^j{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{{\sum }_{s=1}^k{x}_s}{\left[\sum _{s=1}^k{x}_s\right]}_{q^{\alpha }}^{n-i+j}{q}^{{\sum }_{s=1}^k\left(h-\alpha -s\right)x_s}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{{\sum }_{s=1}^k{x}_s}{\left[\sum _{s=1}^k{x}_s\right]}_{q^{\alpha }}^{n-i}{q}^{{\sum }_{s=1}^k\left(h-s\right)x_s}{q}^{\alpha \left({\sum }_{s=1}^k{x}_s\right)\left(i-1\right)}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =\sum _{j=0}^{i-1}{\left(q^{\alpha }-1\right)}^j\left(\begin{array}{c}i-1\hfill \\ j\hfill \end{array}\right){Ẽ}_{n-i+j,q,w}^{\left(\alpha \right)}\left(h,k\right).\end{array}$$

(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 ∈ ℕ and n, i ∈ ℤ₊, 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

$$\begin{array}{c}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)\\ =\underset{k-\text{times}}{\underbrace{{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}}}{w}^{x_1+\cdots +x_k}{\left[x+\sum _{i=1}^k{x}_i\right]}_{q^{\alpha }}^n{q}^{-{\sum }_{j=1}^{k}j{x}_j}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right).\end{array}$$

(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.

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}{q}^{{\sum }_{j=1}^k\left(\alpha n-j\right)x_j+\alpha nx}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =\sum _{j=0}^n\left(\begin{array}{c}n\hfill \\ j\hfill \end{array}\right){\left[\alpha \right]}_q^j{\left(q-1\right)}^j{Ẽ}_{j,q,w}^{\left(\alpha \right)}\left(0,k|x\right),\end{array}$$

and

$$\begin{array}{l}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}{q}^{{\sum }_{j=1}^k\left(\alpha n-j\right)x_j+\alpha nx}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =\frac{{\left[2\right]}_q^k{q}^{\alpha nx}}{\left(1+w{q}^{\alpha n}\right)\cdots \left(1+w{q}^{\alpha n-k+1}\right)}.\end{array}$$

(10)

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

Theorem 3. For α ∈ ℕ and n, k ∈ ℤ₊, 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 - aq^k-1).

Let d ∈ ℕ with d ≡ 1 (mod 2). Then we have

$$\begin{array}{l}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}{\left[x+\sum _{j=1}^k{x}_j\right]}_{q^{\alpha }}^n{q}^{-{\sum }_{j=1}^{k}j{x}_j}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =\frac{{\left[d\right]}_{q^{\alpha }}^n}{{\left[d\right]}_{-q}^k}\sum _{a_1,\cdots ,a_k=0}^{d-1}{w}^{a_1+\cdots +a_k}{q}^{-{\sum }_{j=2}^k\left(j-1\right)a_j}{\left(-1\right)}^{{\sum }_{j=1}^k{a}_j}\times \\ {\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{d\left(x_1+\cdots +x_k\right)}{\left[\frac{x+{\sum }_{j=1}^k{a}_j}{d}+\sum _{j=1}^k{x}_j\right]}_{q^{\alpha d}}^n{q}^{-d{\sum }_{j=1}^{k}j{x}_j}d{\mu }_{-q^d}\left(x_1\right)\cdots d{\mu }_{-q^d}\left(x_k\right)\end{array}$$

(11)

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

Theorem 4. For d ∈ ℕ with d ≡ 1 (mod 2), we have

$$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|x\right)\\ =\frac{{\left[d\right]}_{q^{\alpha }}^n}{{\left[d\right]}_{-q}^k}\sum _{a_1,\cdots ,a_k=0}^{d-1}{\left(-w\right)}^{a_1+\cdots +a_k}{q}^{-{\sum }_{j=2}^k\left(j-1\right)a_j}{Ẽ}_{n,q^d,w^d}^{\left(\alpha \right)}\left(0,k|\frac{x+a_1+\cdots +a_k}{d}\right).\end{array}$$

Moreover,

$$\begin{array}{c}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(0,k|dx\right)\\ =\frac{{\left[d\right]}_{q^{\alpha }}^n}{{\left[d\right]}_{-q}^k}\sum _{a_1,\cdots ,a_k=0}^{d-1}{\left(-w\right)}^{a_1+\cdots +a_k}{q}^{-{\sum }_{j=2}^k\left(j-1\right)a_j}{Ẽ}_{n,q^d,w^d}^{\left(\alpha \right)}\left(0,k|x+\frac{a_1+\cdots +a_k}{d}\right).\end{array}$$

By (8), we get

$$\begin{array}{l}{\tilde{E}}_{n,q,w}^{(\alpha )}(0,k|x={\displaystyle \sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right)}{[x]}_{q^{\alpha }}^{n-l}{q}^{\alpha lx}{\tilde{E}}_{l,q,w}^{(\alpha )}(0,k)\\ ={\left({[x]}_{q^{\alpha }}+q^{\alpha x}{\tilde{E}}_{q,w}^{(\alpha )}(0,k)\right)}^n,\end{array}$$

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

$$\begin{array}{l}{\tilde{E}}_{n,q,w}^{(\alpha )}(0,k|x+y)={\displaystyle \sum _{l=0}^n\left(\begin{array}{c}n\\ l\end{array}\right)}{[y]}_{q^{\alpha }}^{n-l}{q}^{\alpha ly}{\tilde{E}}_{l,q,w}^{(\alpha )}(0,k|x)\\ ={\left({[y]}_{q^{\alpha }}+q^{\alpha y}{\tilde{E}}_{q,w}^{(\alpha )}(0,k|x)\right)}^n.\end{array}$$

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:

$${Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)={\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right).$$

(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 ℤ _p , we see that

$$\begin{array}{c}{q}^{\alpha x}{\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right)\\ =\left(q^{\alpha }-1\right){\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1\right]}_{q^{\alpha }}^{n+1}{q}^{x_1\left(h-\alpha -1\right)}d{\mu }_{-q}\left(x_1\right)+{\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-\alpha -1\right)}d{\mu }_{-q}\left(x_1\right).\end{array}$$

(13)

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

Theorem 5. For α ∈ ℕ and h ∈ ℤ, 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

$$\begin{array}{ll}\hfill {Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)& ={\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right)\\ =\sum _{l=0}^n\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[x\right]}_{q^{\alpha }}^{n-1}{q}^{\alpha lx}{\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x_1\right]}_{q^{\alpha }}^l{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right)\\ =\sum _{l=0}^n\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[x\right]}_{q^{\alpha }}^{n-1}{q}^{\alpha lx}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(h,1\right)\\ ={\left(q^{\alpha x}{Ẽ}_{q,w}^{\left(\alpha \right)}\left(h,1\right)+{\left[x\right]}_{q^{\alpha }}\right)}^n,\text{for}n\ge 1,\end{array}$$

(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

$$\begin{array}{l}w{q}^h{\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1+1\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right)+{\int }_{{\mathbb{Z}}_p}{w}^{x_1}{\left[x+x_1\right]}_{q^{\alpha }}^n{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right)\\ ={\left[2\right]}_q{\left[x\right]}_{q^{\alpha }}^n.\end{array}$$

(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

$$w{q}^h{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|1\right)+{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)=\left\{\begin{array}{cc}{\left[2\right]}_q,\hfill & \text{if}n=0,\hfill \\ 0,\hfill & \text{if}n>0.\hfill \end{array}\right.$$

(17)

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

Theorem 6. For h ∈ ℤ and n ∈ ℤ₊, we have

$$w{q}^h{\left(q^{\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}\left(h,1\right)+1\right)}^n{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)=\left\{\begin{array}{cc}{\left[2\right]}_q,\hfill & \text{if}n=0,\hfill \\ 0,\hfill & \text{if}n>0,\hfill \end{array}\right.$$

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 ℤ _p , we easily get

$${Ẽ}_{0,q,w}^{\left(\alpha \right)}\left(h,1\right)={\int }_{{\mathbb{Z}}_p}{w}^{x_1}{q}^{x_1\left(h-1\right)}d{\mu }_{-q}\left(x_1\right)=\frac{{\left[2\right]}_q}{{\left[2\right]}_{w{q}^h}}.$$

By (12), we see that

$$\begin{array}{l}{Ẽ}_{n,q^{-1},w^{-1}}^{\left(\alpha \right)}\left(h,1|1-x\right)={\int }_{{\mathbb{Z}}_p}{w}^{-1}{\left[1-x+x_1\right]}_{q^{-\alpha }}^n{q}^{-x_1\left(h-1\right)}d{\mu }_{-q^{-1}}\left(x_1\right)\\ ={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}\frac{{\left[2\right]}_q}{{\left(1-q^{\alpha }\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^l{q}^{\alpha lx}\frac{1}{1+w{q}^{\alpha l+h}}\\ ={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|x\right)\end{array}$$

(18)

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

Theorem 7. For α ∈ ℕ, h ∈ ℤ and n ∈ ℤ₊, 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

$$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)\\ ={\left(-1\right)}^{n}w{q}^{\alpha n+h-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1|1\right)\\ ={\left(-1\right)}^{n+1}{q}^{\alpha n-1}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,1\right)\text{if}n\ge 1.\end{array}$$

Let d ∈ ℕ with d ≡ 1 (mod 2). Then we have

$$\begin{array}{l}{\int }_{{\mathbb{Z}}_p}{w}^{x_1}{q}^{x_1\left(h-1\right)}{\left[x+x_1\right]}_{q^{\alpha }}^{n}d{\mu }_{-q}\left(x_1\right)\\ =\frac{{\left[d\right]}_{q^{\alpha }}^n}{{\left[d\right]}_{-q}}\sum _{a=0}^{d-1}{w}^a{q}^{ha}{\left(-1\right)}^a{\int }_{{\mathbb{Z}}_p}{w}^{d{x}_1}{\left[\frac{x+a}{d}+x_1\right]}_{q^{\alpha d}}^n{q}^{x_1\left(h-1\right)d}d{\mu }_{-q^d}\left(x_1\right).\end{array}$$

(19)

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

Theorem 8 (Multiplication formula). For d ∈ ℕ 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

$$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|x\right)\\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}{\left[x_1+\cdots +x_k+x\right]}_{q^{\alpha }}^n{q}^{\left(h-1\right)x_1+\cdots +\left(h-k\right)x_k}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right).\end{array}$$

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

$$\begin{array}{l}w{q}^h{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}{\left[x+1+\sum _{i=1}^k{x}_i\right]}_{q^{\alpha }}^n{q}^{{\sum }_{i=1}^k\left(h-i\right)x_i}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =-{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}{\left[x+\sum _{i=1}^k{x}_i\right]}_{q^{\alpha }}^n{q}^{{\sum }_{i=1}^k\left(h-i\right)x_i}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ +{\left[2\right]}_q{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{x_2+\cdots +x_k}{\left[x+\sum _{i=2}^k{x}_i\right]}_{q^{\alpha }}^n{q}^{{\sum }_{i=2}^k\left(h-i\right)x_i}d{\mu }_{-q}\left(x_2\right)\cdots d{\mu }_{-q}\left(x_k\right).\end{array}$$

(20)

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

Theorem 9. For h ∈ ℤ, α ∈ ℕ and n ∈ ℤ₊, 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

$$\begin{array}{l}{q}^{\alpha x}{\displaystyle {\int }_{{\mathbb{Z}}_p}\cdots }{\displaystyle {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +w_k}}[x+{\displaystyle \sum _{i=1}^k{x}_i{]}_{q^{\alpha }}^n}{q}^{{\displaystyle {\sum }_{i=1}^k(h-i)x_i}}d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_k)\\ =(q^{\alpha }-1){\displaystyle {\int }_{{\mathbb{Z}}_p}\cdots {\displaystyle {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}}}{[x+{\displaystyle \sum _{i=1}^k{x}_i}]}_{q^{\alpha }}^{n+1}{q}^{{\displaystyle {\sum }_{i=1}^k(h-\alpha -i)x_i}}d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_k)\\ +{\displaystyle {\int }_{{\mathbb{Z}}_p}\cdots {\displaystyle {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}}[x+{\displaystyle \sum _{i=1}^k{x}_i{]}_{q^{\alpha }}^n}}{q}^{{\displaystyle {\sum }_{i=1}^k(h-\alpha -i)x_i}}d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_k)\\ =(q^{\alpha }-1){\tilde{E}}_{n+1,q,w}^{(\alpha )}(h-\alpha ,k|x)+{\tilde{E}}_{n,q,w}^{(\alpha )}(h-\alpha ,k|x).\end{array}$$

(21)

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

Theorem 10. For n ∈ ℤ₊, 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 ∈ ℕ with d ≡ 1 (mod 2). Then we get

$$\begin{array}{l}{\displaystyle {\int }_{{\mathbb{Z}}_p}\cdots }{\displaystyle {\int }_{{\mathbb{Z}}_p}{w}^{x_1+\cdots +x_k}}[x+{\displaystyle \sum _{j=1}^k{x}_j{]}_{q^{\alpha }}^n}{q}^{{\displaystyle {\sum }_{j=1}^k(h-j})x_j}d{\mu }_{-q}(x_1)\cdots d{\mu }_{-q}(x_k)\\ =\frac{{[d]}_{q^{\alpha }}^n}{{[d]}_{-q}^k}{\displaystyle \sum _{a_1,\cdots ,a_k=0}^{d-1}{w}^{a_1+\cdots +a_k}}{q}^{h{\displaystyle {\sum }_{j=1}^{k}aj-{\displaystyle {\sum }_{j=2}^k(j-1)a_j}}}{(-1)}^{{\displaystyle {\sum }_{j=1}^k{a}_j}}\times \\ {\displaystyle {\int }_{{\mathbb{Z}}_p}\cdots }{\displaystyle {\int }_{{\mathbb{Z}}_p}{w}^{d(x_1+\cdots +x_k)}}{\left[\frac{x+{\displaystyle {\sum }_{j=1}^k{a}_j}}{d}+{\displaystyle \sum _{j=1}^k{x}_j}\right]}_{q^{\alpha d}}^n{q}^{d{\displaystyle {\sum }_{j=1}^k(h-j)x_j}}d{\mu }_{-q^d}(x_1)\cdots d{\mu }_{-q^d}(x_k).\end{array}$$

(22)

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

Theorem 11. For d ∈ ℕ with d ≡ 1 (mod 2), we have

$$\begin{array}{l}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(h,k|dx\right)\\ =\frac{{\left[d\right]}_{q^{\alpha }}^n}{{\left[d\right]}_{-q}^k}\sum _{a_1,\cdots ,a_k=0}^{d-1}{w}^{a_1+\cdots +a_k}{q}^{h{\sum }_{j=1}^{k}aj-{\sum }_{j=2}^k\left(j-1\right)a_j}{\left(-1\right)}^{{\sum }_{j=1}^k{a}_j}{Ẽ}_{n,q^d,w^d}^{\left(\alpha \right)}\left(h,k|x+\frac{a_1+\cdots +a_k}{d}\right).\end{array}$$

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

$$\begin{array}{l}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{-\left(x_1+\cdots +x_k\right)}{\left[k-x+\sum _{i=1}^k{x}_i\right]}_{q^{-\alpha }}^n{q}^{-{\sum }_{i=1}^k\left(k-i\right)x_i}d{\mu }_{-q^{-1}}\left(x_1\right)\cdots d{\mu }_{-q^{-1}}\left(x_k\right)\\ =\frac{q^{\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}}{{\left(1-q^{-\alpha }\right)}^n}{\left[2\right]}_q^k\sum _{l=0}^n\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^l{q}^{\alpha lx}\frac{1}{\left(1+w{q}^{\alpha l+1}\right)\cdots \left(1+w{q}^{\alpha l+k}\right)}\\ ={\left(-1\right)}^n{q}^{n\alpha }{q}^{\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}\frac{{\left[2\right]}_q^k}{{\left(1-q^{\alpha }\right)}^n}\sum _{l=0}^n\frac{\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(-1\right)}^l{q}^{\alpha lx}}{\left(1+w{q}^{\alpha l+1}\right)\cdots \left(1+w{q}^{\alpha l+k}\right)}\\ ={\left(-1\right)}^n{q}^{\alpha n+\left(\begin{array}{c}k\hfill \\ 2\hfill \end{array}\right)}{Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right).\end{array}$$

(23)

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

Theorem 12. For n ∈ ℤ₊, 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

$$\begin{array}{l}\sum _{l=0}^n\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left(q^{\alpha }-1\right)}^l{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{{\sum }_{i=1}^k{x}_k}{\left[\sum _{i=1}^k{x}_k\right]}_{q^{\alpha }}^l{q}^{{\sum }_{l=i}^k\left(k-i\right)x_i}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =\frac{{\left[2\right]}_q^k}{\left(1+w{q}^{\alpha n+k}\right)\left(1+w{q}^{\alpha n+k-1}\right)\cdots \left(1+w{q}^{\alpha n+1}\right)}.\end{array}$$

(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

$$\begin{array}{ll}\hfill {Ẽ}_{n,q,w}^{\left(\alpha \right)}\left(k|x\right)& ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{w}^{{\sum }_{i=1}^k{x}_k}{\left[x+\sum _{i=1}^k{x}_k\right]}_{q^{\alpha }}^n{q}^{{\sum }_{i=1}^k\left(k-i\right)x_i}d{\mu }_{-q}\left(x_1\right)\cdots d{\mu }_{-q}\left(x_k\right)\\ =\sum _{l=0}^n\left(\begin{array}{c}n\hfill \\ l\hfill \end{array}\right){\left[x\right]}_{q^{\alpha }}^{n-l}{q}^{\alpha lx}{Ẽ}_{l,q,w}^{\left(\alpha \right)}\left(k|0\right)\\ ={\left(q^{x\alpha }{Ẽ}_{q,w}^{\left(\alpha \right)}\left(k|0\right)+{\left[x\right]}_{q^{\alpha }}\right)}^n,n\in {\mathbb{Z}}_{+},\end{array}$$

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