On the modified q-Euler polynomials with weight

Abstract

In this paper, we construct a new q-extension of Euler numbers and polynomials with weight related to fermionic p-adic q-integral on ${\mathbb{Z}}_p$ and give new explicit formulas related to these numbers and polynomials.

Throughout this paper ${\mathbb{Z}}_p$, ${\mathbb{Q}}_p$ and ${\mathbb{C}}_p$ will respectively denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of ${\mathbb{Q}}_p$. Let ${\nu }_p$ be the normalized exponential valuation of ${\mathbb{C}}_p$ with ${|p|}_p=p^{-{\nu }_p(p)}=\frac{1}{p}$.

In this paper, we assume that $q\in {\mathbb{C}}_p$ with ${|1-q|}_p<p^{-\frac{1}{p-1}}$ so that $q^x=exp(xlogq)$ for $x\in {\mathbb{Z}}_p$. The q-number of x is denoted by ${[x]}_q=\frac{1-q^x}{1-q}$. Note that ${lim}_{q\to 1}{[x]}_q=x$. Let d be a fixed integer bigger than 0, and let p be a fixed prime number and $(d,p)=1$. We set

$$\begin{array}{c}{X}_d=\underset{\overleftarrow{N}}{lim}\mathbb{Z}/d{p}^N\mathbb{Z},X^{\ast }=\underset{(a,p)=1}{\bigcup _{0<a<dp}}(a+dp{\mathbb{Z}}_p),\hfill \\ a+d{p}^N{\mathbb{Z}}_p=\{x\in X\mid x\equiv a(modd{p}^N)\},\hfill \end{array}$$

where $a\in \mathbb{Z}$ lies in $0\le a<d{p}^N$ (see [1–22]).

Let $C({\mathbb{Z}}_p)$ be the space of continuous functions on ${\mathbb{Z}}_p$. For $f\in C({\mathbb{Z}}_p)$, the fermionic p-adic q-integral on ${\mathbb{Z}}_p$ is defined by Kim as

$$I_q(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_{-q}(x)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{{[p^N]}_q}\sum _{x=0}^{p^N-1}f(x){(-q)}^x\text{(see [8–22])}.$$

As is well known, Euler polynomials are defined by the generating function to be

$$\frac{2}{e^t+1}{e}^{xt}=e^{E(x)t}=\sum _{n=0}^{\mathrm{\infty }}{E}_n(x)\frac{t^n}{n!}\text{(see [11–13, 15, 20–22])}$$

with the usual convention about replacing $E^n(x)$ by $E_n(x)$. In the special case, $x=0$, $E_n(0)=E_n$ are called the nth Euler numbers.

In [13, 20, 23], Kim defined the q-Euler numbers as follows:

$$E_{0,q}=1,q{(qE+1)}^n+E_{n,q}=\{\begin{array}{cc}{[2]}_q,\hfill & \text{if }n=0,\hfill \\ 0,\hfill & \text{if }n\ne 0,\hfill \end{array}$$

(1)

with the usual convection of replacing $E^n$ by $E_{n,q}$. From (1), we also derive

$$E_{n,q}=\frac{{[2]}_q}{{(1-q)}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{(-1)}^l}{1+q^{l+1}}\text{(see [20, 23])}.$$

By using an invariant p-adic q-integral on ${\mathbb{Z}}_p$, a q-extension of ordinary Euler polynomials, called q-Euler polynomials, is considered and investigated by Kim [14, 15, 18]. For $x\in {\mathbb{Z}}_p$, q-Euler polynomials are defined as follows:

$$E_{n,q}(x)={\int }_{{\mathbb{Z}}_p}{[x+y]}_q^{n}d{\mu }_{-q}(y).$$

(2)

By (2), the following relation holds:

$$E_{n,q}(x)=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right){[x]}_q^{n-k}{q}^{kx}{E}_{k,q}.$$

Recently, Kim considered the modified q-Euler polynomials which are slightly different from Kim’s q-Euler polynomials as follows:

$${ϵ}_{n,q}(x)={\int }_{{\mathbb{Z}}_p}{q}^{-x}{[x+y]}_q^{n}d{\mu }_{-q}(y)\text{for }n\in \mathbb{N},$$

and he showed that

$${ϵ}_{n,q}(x)=\frac{{[2]}_q}{{(1-q)}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{q^{xl}}{1+q^l}$$

(3)

(see [22]). In the special case, $x=0$, ${ϵ}_{n,q}(0)={ϵ}_{n,q}$ are called the nth modified q-Euler numbers, and it is showed that

$${ϵ}_{n,q}=\frac{{[2]}_q}{{(1-q)}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{1}{1+q^l}.$$

(4)

And in [24], authors defined modified q-Euler polynomials with weight α ${ϵ}_{n,q}^{(\alpha )}(x)$ as follows:

$${ϵ}_{n,q}^{(\alpha )}(x)={\int }_{{\mathbb{Z}}_p}{q}^{-x}{[x+y]}_{q^{\alpha }}^{n}d{\mu }_{-q^{\alpha }}(y)$$

and proved that

$${ϵ}_{n,q}^{(\alpha )}(x)=\frac{{[2]}_q}{{(1-q^{\alpha })}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){(-1)}^l\frac{q^{\alpha l}}{1+q^{\alpha l}}.$$

(5)

In the special case, $x=0$, ${ϵ}_{n,q}^{(\alpha )}(0)={ϵ}_{n,q}^{(\alpha )}$ are called the nth modified q-Euler numbers with weight α, and it is showed that

$$\begin{array}{rcl}{ϵ}_{n,q}^{(\alpha )}& =& \frac{{[2]}_q}{{(1-q^{\alpha })}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){(-1)}^l{q}^{\alpha l}\frac{1}{1+q^{\alpha l}}\\ =& {[2]}_q\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m{[m+x]}_{q^{\alpha }}^n.\end{array}$$

(6)

In this paper, we construct a new q-extension of Euler numbers and polynomials with weight related to fermionic p-adic q-integral on ${\mathbb{Z}}_p$ and give new explicit formulas related to these numbers and polynomials.

1 A new approach of modified q-Euler polynomials

Let us consider the following modified q-Euler numbers:

$$\begin{array}{rcl}{\widetilde{ϵ}}_{n,q}(x)& =& {\int }_{{\mathbb{Z}}_p}{q}^{-y}{(x+{[y]}_q)}^{n}d{\mu }_{-q}(y)\\ =& \sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)x^{n-l}{ϵ}_{l,q}=\sum _{l=0}^n\sum _{k=0}^l\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{l}{k}\right)\frac{{[2]}_q}{{(1-q)}^l}\frac{x^{n-l}}{1+q^k},\end{array}$$

where

$${\widetilde{ϵ}}_{n,q}(0)={ϵ}_{n,q}=\frac{{[2]}_q}{{(1-q)}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{1}{1+q^l}.$$

(7)

Thus, by (7),

$${(1-q)}^n{ϵ}_{n,q}={[2]}_q\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{1}{1+q^l}.$$

Consider the equation

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{(1-q)}^n{ϵ}_{n,q}\frac{t^n}{n!}& =& {[2]}_q\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{1}{1+q^l}\frac{t^n}{n!}={[2]}_q\left(\sum _{m=0}^{\mathrm{\infty }}\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+q^l}\frac{t^l}{l!}\right)\\ =& {[2]}_q{e}^t\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+q^l}\frac{t^l}{l!}\right).\end{array}$$

Since

$$\begin{array}{rcl}{e}^{(1-q)xt}\sum _{n=0}^{\mathrm{\infty }}{(1-q)}^n{ϵ}_{n,q}\frac{t^n}{n!}& =& \left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(1-q)}^l{x}^l{t}^l}{l!}\right)\left(\sum _{n=0}^{\mathrm{\infty }}{(1-q)}^n{ϵ}_{n,q}\frac{t^n}{n!}\right)\\ =& \sum _{m=0}^{\mathrm{\infty }}{(1-q)}^m\sum _{n=0}^m\left(\genfrac{}{}{0ex}{}{m}{n}\right){ϵ}_{n,q}{x}^{m-n}\frac{t^m}{m!}\\ =& \sum _{m=0}^{\mathrm{\infty }}{(1-q)}^m{\widetilde{ϵ}}_{m,q}(x)\frac{t^m}{m!}\end{array}$$

(8)

and

$$\begin{array}{rcl}{e}^{(1-q)xt}{[2]}_q{e}^t\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+q^l}\frac{t^l}{l!}\right)& =& {[2]}_q{e}^{((1-q)x+1)t}\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+q^l}\frac{t^l}{l!}\right)\\ =& {[2]}_q\left(\sum _{m=0}^{\mathrm{\infty }}{((1-q)x+1)}^m\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+q^l}\frac{t^l}{l!}\right)\\ =& {[2]}_q\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{((1-q)x+1)}^{n-l}}{1+q^l}\frac{t^n}{n!},\end{array}$$

(9)

by (8) and (9), we get

$$\begin{array}{rcl}{(1-q)}^n{\widetilde{ϵ}}_{n,q}(x)& =& {[2]}_q\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{((1-q)x+1)}^{n-l}}{1+q^l}\\ =& {[2]}_q\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{1}{1+q^l}\sum _{j=0}^{n-l}\left(\genfrac{}{}{0ex}{}{n-l}{j}\right){(1-q)}^j{x}^j.\end{array}$$

Thus, we have the following result.

Theorem 1.1 For $n\ge 1$,

$$\begin{array}{rcl}{\widetilde{ϵ}}_{n,q}(x)& =& \frac{{[2]}_q}{{(1-q)}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{((1-q)x+1)}^{n-l}}{1+q^l}\\ =& \frac{{[2]}_q}{{(1-q)}^n}\sum _{l=0}^n\sum _{j=0}^{n-l}\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{n-l}{j}\right)\frac{{(1-q)}^j}{1+q^l}{x}^j.\end{array}$$

2 A new approach of q-Euler polynomials with weight α

Let us consider the following modified q-Euler polynomials with weight α:

$$\begin{array}{rcl}{\widetilde{ϵ}}_{n,q}^{(\alpha )}(x)& =& {\int }_{{\mathbb{Z}}_p}{q}^{-y}{(x+{[y]}_{q^{\alpha }})}^{n}d{\mu }_{-q^{\alpha }}(y)\\ =& \sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)x^{n-l}{ϵ}_{k,q}^{(\alpha )}=\sum _{k=0}^n\sum _{l=0}^k\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{k}{l}\right)\frac{{[2]}_{q^{\alpha }}}{{(1-q)}^n}\frac{{(-1)}^l}{1+q^{\alpha +l}}{x}^{n-k},\end{array}$$

where

$${\widetilde{ϵ}}_{n,q}^{(\alpha )}(0)={ϵ}_{n,q}^{(\alpha )}=\frac{{[2]}_q}{{(1-q^{\alpha })}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{(-1)}^l{q}^{\alpha l}}{1+q^{\alpha l}}.$$

(10)

Thus, by (10), we have

$${(1-q^{\alpha })}^n{ϵ}_{n,q}^{(\alpha )}={[2]}_q\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{(-1)}^l{q}^{\alpha l}}{1+q^{\alpha l}}.$$

Consider the equation

$$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{(1-q^{\alpha })}^n{ϵ}_{n,q}^{(\alpha )}\frac{t^n}{n!}& =& {[2]}_q\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{(-1)}^l{q}^{\alpha l}}{1+q^{\alpha l}}\frac{t^n}{n!}={[2]}_q\left(\sum _{m=0}^{\mathrm{\infty }}\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(-1)}^l{q}^{\alpha l}}{1+q^{\alpha l}}\frac{t^l}{l!}\right)\\ =& {[2]}_q{e}^t\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(-q^{\alpha })}^l}{1+q^{\alpha l}}\frac{t^l}{l!}\right).\end{array}$$

Since

$$\begin{array}{rcl}{e}^{(1-q^{\alpha })xt}\sum _{n=0}^{\mathrm{\infty }}{(1-q^{\alpha })}^n{ϵ}_{n,q}^{(\alpha )}\frac{t^n}{n!}& =& \left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(1-q^{\alpha })}^l{x}^l{t}^l}{l!}\right)\left(\sum _{n=0}^{\mathrm{\infty }}{(1-q^{\alpha })}^n{ϵ}_{n,q}^{(\alpha )}\frac{t^n}{n!}\right)\\ =& \sum _{m=0}^{\mathrm{\infty }}{(1-q^{\alpha })}^m\sum _{n=0}^m\left(\genfrac{}{}{0ex}{}{m}{n}\right){ϵ}_{n,q}^{(\alpha )}{x}^{m-n}\frac{t^m}{m!}\\ =& \sum _{m=0}^{\mathrm{\infty }}{(1-q^{\alpha })}^m{\widetilde{ϵ}}_{m,q}^{(\alpha )}(x)\frac{t^m}{m!}\end{array}$$

(11)

and

$$\begin{array}{rcl}{e}^{(1-q^{\alpha })xt}{[2]}_q{e}^t\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(-q^{\alpha })}^l}{1+q^{\alpha l}}\frac{t^l}{l!}\right)& =& {[2]}_q{e}^{((1-q^{\alpha })x+1)t}\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(-q^{\alpha })}^l}{1+q^{\alpha l}}\frac{t^l}{l!}\right)\\ =& {[2]}_q\left(\sum _{m=0}^{\mathrm{\infty }}{((1-q^{\alpha })x+1)}^m\frac{t^m}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{(-q^{\alpha })}^l}{1+q^{\alpha l}}\frac{t^l}{l!}\right)\\ =& {[2]}_q\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{((1-q^{\alpha })x+1)}^{n-l}}{1+q^{\alpha l}}{(-q^{\alpha })}^l\frac{t^n}{n!},\end{array}$$

(12)

by (11) and (12), we get

$$\begin{array}{rcl}{(1-q^{\alpha })}^n{\widetilde{ϵ}}_{n,q}^{(\alpha )}(x)& =& {[2]}_q\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\frac{{((1-q^{\alpha })x+1)}^{n-l}}{1+q^{\alpha l}}{(-q^{\alpha })}^l\\ =& {[2]}_q\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{(-q^{\alpha })}^l}{1+q^{\alpha l}}\sum _{j=0}^{n-l}\left(\genfrac{}{}{0ex}{}{n-l}{j}\right){(1-q^{\alpha })}^j{x}^j.\end{array}$$

Thus, we have the following result.

Theorem 2.1 For $n\ge 1$,

$$\begin{array}{rcl}{\widetilde{ϵ}}_{n,q}^{(\alpha )}(x)& =& \frac{{[2]}_q}{{(1-q^{\alpha })}^n}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right)\frac{{(-q^{\alpha })}^l{((1-q^{\alpha })x+1)}^{n-l}}{1+q^{\alpha l}}\\ =& \frac{{[2]}_q}{{(1-q^{\alpha })}^n}\sum _{l=0}^n\sum _{j=0}^{n-l}\left(\genfrac{}{}{0ex}{}{n}{l}\right)\left(\genfrac{}{}{0ex}{}{n-l}{j}\right)\frac{{(-q^{\alpha })}^l{(1-q^{\alpha })}^j}{1+q^{\alpha l}}{x}^j.\end{array}$$

A systemic study of some families of the modified q-Euler polynomials with weight is presented by using the multivariate fermionic p-adic integral on ${\mathbb{Z}}_p$. The study of these modified q-Euler numbers and polynomials yields an interesting q-analogue of identities for Stirling numbers.

In recent years, many mathematicians and physicists have investigated zeta functions, multiple zeta functions, L-functions, and multiple q-Bernoulli numbers and polynomials mainly because of their interest and importance. These functions and polynomials are used not only in complex analysis and mathematical physics, but also in p-adic analysis and other areas. In particular, multiple zeta functions and multiple L-functions occur within the context of knot theory, quantum field theory, applied analysis and number theory (see [1–29]).

In our subsequent papers, we shall apply this p-adic mathematical theory to quantum statistical mechanics. Using p-adic quantum statistical mechanics, we can also derive a new partition function in the p-adic space and adopt this new partition function to quantum transport theory which is based on the projection technique related to the Liouville equation. We expect that a new quantum transport theory will explain diverse physical properties of the condensed matter system.