Theory and Modern Applications

# 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 ${|p|}_{p}={p}^{-{\nu }_{p}\left(p\right)}=\frac{1}{p}$.

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

$\begin{array}{c}{X}_{d}=\underset{\stackrel{←}{N}}{lim}\mathbb{Z}/d{p}^{N}\mathbb{Z},\phantom{\rule{2em}{0ex}}{X}^{\ast }=\underset{\left(a,p\right)=1}{\bigcup _{0

where $a\in \mathbb{Z}$ lies in $0\le a (see ).

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

${I}_{q}\left(f\right)={\int }_{{\mathbb{Z}}_{p}}f\left(x\right)\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(x\right)=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{{\left[{p}^{N}\right]}_{q}}\sum _{x=0}^{{p}^{N}-1}f\left(x\right){\left(-q\right)}^{x}\phantom{\rule{1em}{0ex}}\text{(see [8–22])}.$

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

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

with the usual convention about replacing ${E}^{n}\left(x\right)$ by ${E}_{n}\left(x\right)$. In the special case, $x=0$, ${E}_{n}\left(0\right)={E}_{n}$ are called the nth Euler numbers.

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

(1)

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

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

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:

${E}_{n,q}\left(x\right)={\int }_{{\mathbb{Z}}_{p}}{\left[x+y\right]}_{q}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(y\right).$
(2)

By (2), the following relation holds:

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

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

and he showed that

${ϵ}_{n,q}\left(x\right)=\frac{{\left[2\right]}_{q}}{{\left(1-q\right)}^{n}}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right)\frac{{q}^{xl}}{1+{q}^{l}}$
(3)

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

${ϵ}_{n,q}=\frac{{\left[2\right]}_{q}}{{\left(1-q\right)}^{n}}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right)\frac{1}{1+{q}^{l}}.$
(4)

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

${ϵ}_{n,q}^{\left(\alpha \right)}\left(x\right)={\int }_{{\mathbb{Z}}_{p}}{q}^{-x}{\left[x+y\right]}_{{q}^{\alpha }}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-{q}^{\alpha }}\left(y\right)$

and proved that

${ϵ}_{n,q}^{\left(\alpha \right)}\left(x\right)=\frac{{\left[2\right]}_{q}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){\left(-1\right)}^{l}\frac{{q}^{\alpha l}}{1+{q}^{\alpha l}}.$
(5)

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

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

$\begin{array}{rcl}{\stackrel{˜}{ϵ}}_{n,q}\left(x\right)& =& {\int }_{{\mathbb{Z}}_{p}}{q}^{-y}{\left(x+{\left[y\right]}_{q}\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-q}\left(y\right)\\ =& \sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right){x}^{n-l}{ϵ}_{l,q}=\sum _{l=0}^{n}\sum _{k=0}^{l}\left(\genfrac{}{}{0}{}{n}{l}\right)\left(\genfrac{}{}{0}{}{l}{k}\right)\frac{{\left[2\right]}_{q}}{{\left(1-q\right)}^{l}}\frac{{x}^{n-l}}{1+{q}^{k}},\end{array}$

where

${\stackrel{˜}{ϵ}}_{n,q}\left(0\right)={ϵ}_{n,q}=\frac{{\left[2\right]}_{q}}{{\left(1-q\right)}^{n}}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right)\frac{1}{1+{q}^{l}}.$
(7)

Thus, by (7),

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

Consider the equation

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

Since

$\begin{array}{rcl}{e}^{\left(1-q\right)xt}\sum _{n=0}^{\mathrm{\infty }}{\left(1-q\right)}^{n}{ϵ}_{n,q}\frac{{t}^{n}}{n!}& =& \left(\sum _{l=0}^{\mathrm{\infty }}\frac{{\left(1-q\right)}^{l}{x}^{l}{t}^{l}}{l!}\right)\left(\sum _{n=0}^{\mathrm{\infty }}{\left(1-q\right)}^{n}{ϵ}_{n,q}\frac{{t}^{n}}{n!}\right)\\ =& \sum _{m=0}^{\mathrm{\infty }}{\left(1-q\right)}^{m}\sum _{n=0}^{m}\left(\genfrac{}{}{0}{}{m}{n}\right){ϵ}_{n,q}{x}^{m-n}\frac{{t}^{m}}{m!}\\ =& \sum _{m=0}^{\mathrm{\infty }}{\left(1-q\right)}^{m}{\stackrel{˜}{ϵ}}_{m,q}\left(x\right)\frac{{t}^{m}}{m!}\end{array}$
(8)

and

$\begin{array}{rcl}{e}^{\left(1-q\right)xt}{\left[2\right]}_{q}{e}^{t}\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+{q}^{l}}\frac{{t}^{l}}{l!}\right)& =& {\left[2\right]}_{q}{e}^{\left(\left(1-q\right)x+1\right)t}\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+{q}^{l}}\frac{{t}^{l}}{l!}\right)\\ =& {\left[2\right]}_{q}\left(\sum _{m=0}^{\mathrm{\infty }}{\left(\left(1-q\right)x+1\right)}^{m}\frac{{t}^{m}}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}\frac{1}{1+{q}^{l}}\frac{{t}^{l}}{l!}\right)\\ =& {\left[2\right]}_{q}\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right)\frac{{\left(\left(1-q\right)x+1\right)}^{n-l}}{1+{q}^{l}}\frac{{t}^{n}}{n!},\end{array}$
(9)

by (8) and (9), we get

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

Thus, we have the following result.

Theorem 1.1 For $n\ge 1$,

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

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

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

$\begin{array}{rcl}{\stackrel{˜}{ϵ}}_{n,q}^{\left(\alpha \right)}\left(x\right)& =& {\int }_{{\mathbb{Z}}_{p}}{q}^{-y}{\left(x+{\left[y\right]}_{{q}^{\alpha }}\right)}^{n}\phantom{\rule{0.2em}{0ex}}d{\mu }_{-{q}^{\alpha }}\left(y\right)\\ =& \sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){x}^{n-l}{ϵ}_{k,q}^{\left(\alpha \right)}=\sum _{k=0}^{n}\sum _{l=0}^{k}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{k}{l}\right)\frac{{\left[2\right]}_{{q}^{\alpha }}}{{\left(1-q\right)}^{n}}\frac{{\left(-1\right)}^{l}}{1+{q}^{\alpha +l}}{x}^{n-k},\end{array}$

where

${\stackrel{˜}{ϵ}}_{n,q}^{\left(\alpha \right)}\left(0\right)={ϵ}_{n,q}^{\left(\alpha \right)}=\frac{{\left[2\right]}_{q}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right)\frac{{\left(-1\right)}^{l}{q}^{\alpha l}}{1+{q}^{\alpha l}}.$
(10)

Thus, by (10), we have

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

Consider the equation

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

Since

$\begin{array}{rcl}{e}^{\left(1-{q}^{\alpha }\right)xt}\sum _{n=0}^{\mathrm{\infty }}{\left(1-{q}^{\alpha }\right)}^{n}{ϵ}_{n,q}^{\left(\alpha \right)}\frac{{t}^{n}}{n!}& =& \left(\sum _{l=0}^{\mathrm{\infty }}\frac{{\left(1-{q}^{\alpha }\right)}^{l}{x}^{l}{t}^{l}}{l!}\right)\left(\sum _{n=0}^{\mathrm{\infty }}{\left(1-{q}^{\alpha }\right)}^{n}{ϵ}_{n,q}^{\left(\alpha \right)}\frac{{t}^{n}}{n!}\right)\\ =& \sum _{m=0}^{\mathrm{\infty }}{\left(1-{q}^{\alpha }\right)}^{m}\sum _{n=0}^{m}\left(\genfrac{}{}{0}{}{m}{n}\right){ϵ}_{n,q}^{\left(\alpha \right)}{x}^{m-n}\frac{{t}^{m}}{m!}\\ =& \sum _{m=0}^{\mathrm{\infty }}{\left(1-{q}^{\alpha }\right)}^{m}{\stackrel{˜}{ϵ}}_{m,q}^{\left(\alpha \right)}\left(x\right)\frac{{t}^{m}}{m!}\end{array}$
(11)

and

$\begin{array}{rcl}{e}^{\left(1-{q}^{\alpha }\right)xt}{\left[2\right]}_{q}{e}^{t}\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{\left(-{q}^{\alpha }\right)}^{l}}{1+{q}^{\alpha l}}\frac{{t}^{l}}{l!}\right)& =& {\left[2\right]}_{q}{e}^{\left(\left(1-{q}^{\alpha }\right)x+1\right)t}\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{\left(-{q}^{\alpha }\right)}^{l}}{1+{q}^{\alpha l}}\frac{{t}^{l}}{l!}\right)\\ =& {\left[2\right]}_{q}\left(\sum _{m=0}^{\mathrm{\infty }}{\left(\left(1-{q}^{\alpha }\right)x+1\right)}^{m}\frac{{t}^{m}}{m!}\right)\left(\sum _{l=0}^{\mathrm{\infty }}\frac{{\left(-{q}^{\alpha }\right)}^{l}}{1+{q}^{\alpha l}}\frac{{t}^{l}}{l!}\right)\\ =& {\left[2\right]}_{q}\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^{n}\left(\genfrac{}{}{0}{}{n}{l}\right)\frac{{\left(\left(1-{q}^{\alpha }\right)x+1\right)}^{n-l}}{1+{q}^{\alpha l}}{\left(-{q}^{\alpha }\right)}^{l}\frac{{t}^{n}}{n!},\end{array}$
(12)

by (11) and (12), we get

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

Thus, we have the following result.

Theorem 2.1 For $n\ge 1$,

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

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 ).

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.

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## Acknowledgements

The authors are grateful for the valuable comments and suggestions of the referees.

## Author information

Authors

### Corresponding author

Correspondence to Jin-Woo Park.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Rim, SH., Park, JW., Kwon, J. et al. On the modified q-Euler polynomials with weight. Adv Differ Equ 2013, 356 (2013). https://doi.org/10.1186/1687-1847-2013-356

• fermionic p-adic q-integral on ${\mathbb{Z}}_{p}$ 