Keywords

2010 Mathematics Subject Classification

1 Introduction and Preliminaries

Let p be a fixed odd prime. Throughout this paper, \(\mathbb {Z}_{p},\mathbb {Q}_{p}\) and \(\mathbb {C}_{p}\) will be the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of \(\mathbb {Q}_{p}\), respectively.

The p-adic norm \(\left| \cdot \right| _{p}\) in \(\mathbb {C}_{p}\) is normalized as \(\left| p\right| _{p}=\frac{1}{p}\). Let \(f\left( x\right) \) be continuous function on \(\mathbb {Z}_{p}\). Then the fermionic p-adic integral on \(\mathbb {Z}_{p}\) is defined as

$$\begin{aligned} I_{-1}\left( f\right)&=\int _{\mathbb {Z}_{p}}f\left( x\right) d\mu _{-1}\left( x\right) \\&=\lim _{N\rightarrow \infty }\sum _{x=0}^{p^{N}-1}f\left( x\right) \left( -1\right) ^{x},\quad \left( \text {see [9]}\right) .\nonumber \end{aligned}$$
(1.1)

From (1.1), we note that

$$\begin{aligned} I_{-1}\left( f_{n}\right) +\left( -1\right) ^{n-1}I_{-1}\left( f\right) =2\sum _{l=0}^{n-1}\left( -1\right) ^{n-1-l}f\left( l\right) ,\quad \left( \text {see [7]}\right) , \end{aligned}$$
(1.2)

where \(n\in \mathbb {N}\).

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

$$\begin{aligned} \int _{\mathbb {Z}_{p}}e^{\left( x+y\right) t}d\mu _{-1}\left( y\right) =\frac{2}{e^{t}+1}e^{xt}=\sum _{n=0}^{\infty }E_{n}\left( x\right) \frac{t^{n}}{n!}. \end{aligned}$$
(1.3)

When \(x=0\), \(E_{n}=E_{n}\left( 0\right) \) are called the Euler numbers (see [1–19]).

For a fixed odd integer d with \(\left( p,d\right) =1\), we set

where \(a\in \mathbb {Z}\) lies in \(0\le a<dp^{N}\).

It is known that

$$ \int _{\mathbb {Z}_{p}}f\left( x\right) d\mu _{-1}\left( x\right) =\int _{X}f\left( x\right) d\mu _{-1}\left( x\right) ,\quad \left( \text {see}\ [7{-}9]\right) , $$

where f is a continuous function on \(\mathbb {Z}_{p}\).

Let \(d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\) and let \(\chi \) be a Dirichlet character with conductor d. Then the generalized Euler polynomials attached to \(\chi \) are defined by the generating function

$$\begin{aligned} \left( \frac{2\sum _{a=0}^{d-1}\left( -1\right) ^{a}\chi \left( a\right) e^{at}}{e^{dt}+1}\right) e^{xt}=\sum _{n=0}^{\infty }E_{n,\chi }\left( x\right) \frac{t^{n}}{n!}. \end{aligned}$$
(1.4)

In particular, for \(x=0\), \(E_{n,\chi }=E_{n,\chi }\left( 0\right) \) are called the generalized Euler numbers attached to \(\chi \).

For \(d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\), by (1.2), we get

$$\begin{aligned}&\int _{X}\chi \left( y\right) e^{\left( x+y\right) t}d\mu _{-1}\left( y\right) \\&\quad =\frac{2\sum _{a=0}^{d-1}\left( -1\right) ^{a}\chi \left( a\right) e^{at}}{e^{dt}+1}e^{xt}\nonumber \\&\quad =\sum _{n=0}^{\infty }E_{n,\chi }\left( x\right) \frac{t^{n}}{n!},\quad \left( \text {see}\ [9{-}11]\right) .\nonumber \end{aligned}$$
(1.5)

From (1.5), we have

$$\begin{aligned} \int _{X}\chi \left( y\right) \left( x+y\right) ^{n}d\mu _{-1}\left( y\right) =E_{n,\chi }\left( x\right) ,\quad \left( n\ge 0\right) . \end{aligned}$$
(1.6)

Carlitz considered the degenerate Euler polynomials given by the generating function

$$\begin{aligned}&\frac{2}{\left( 1+\lambda t\right) ^{\frac{1}{\lambda }}+1}\left( 1+\lambda t\right) ^{\frac{x}{\lambda }}\\&\quad =\sum _{n=0}^{\infty }\mathcal {E}_{n}\left( x\mid \lambda \right) \frac{t^{n}}{n!},\quad \left( \text {see [3]}\right) .\nonumber \end{aligned}$$
(1.7)

Note that \(\lim _{\lambda \rightarrow 0}\mathcal {E}_{n}\left( x\mid \lambda \right) =E_{n}\left( x\right) \), \(\left( n\ge 0\right) \).

From (1.2), we note that

$$\begin{aligned}&\int _{X}\left( 1+\lambda t\right) ^{\frac{x+y}{\lambda }}d\mu _{-1}\left( y\right) \\&\quad =\frac{2}{\left( 1+\lambda t\right) ^{\frac{1}{\lambda }}+1}\left( 1+\lambda t\right) ^{\frac{x}{\lambda }}\nonumber \\&\quad =\sum _{n=0}^{\infty }\mathcal {E}_{n}\left( x\mid \lambda \right) \frac{t^{n}}{n!}.\nonumber \end{aligned}$$
(1.8)

Thus, by (1.8), we get

$$\begin{aligned} \int _{X}\left( y+x\mid \lambda \right) _{n}d\mu _{-1}\left( y\right) =\mathcal {E}_{n}\left( x\mid \lambda \right) ,\quad \left( n\ge 0\right) , \end{aligned}$$
(1.9)

where \(\left( x\mid \lambda \right) _{n}=x\left( x-\lambda \right) \cdots \left( x-\left( n-1\right) \lambda \right) \), for \(n\ge 1\), and \(\left( x\mid \lambda \right) _{0}=1\).

From (1.2), we can derive the following equation:

$$\begin{aligned}&\int _{X}\chi \left( y\right) \left( 1+\lambda t\right) ^{\frac{x+y}{\lambda }}d\mu _{-1}\left( y\right) \\&\quad =\frac{2\sum _{a=0}^{d-1}\left( -1\right) ^{a}\chi \left( a\right) \left( 1+\lambda t\right) ^{\frac{a}{\lambda }}}{\left( 1+\lambda t\right) ^{\frac{d}{\lambda }}+1}\left( 1+\lambda t\right) ^{\frac{x}{\lambda }},\nonumber \end{aligned}$$
(1.10)

where \(d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\).

In view of (1.5), we define the generalized degenerate Euler polynomials attached to \(\chi \) as follows:

$$\begin{aligned} \frac{2\sum _{a=0}^{d-1}\left( -1\right) ^{a}\chi \left( a\right) \left( 1+\lambda t\right) ^{\frac{a}{\lambda }}}{\left( 1+\lambda t\right) ^{\frac{d}{\lambda }}+1}\left( 1+\lambda t\right) ^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }\mathcal {E}_{n,\lambda ,\chi }\left( x\right) \frac{t^{n}}{n!}. \end{aligned}$$
(1.11)

When \(x=0\), \(\mathcal {E}_{n,\lambda ,\chi }=\mathcal {E}_{n,\lambda ,\chi }\left( 0\right) \) are called the generalized degenerate Euler numbers attached to \(\chi \).

Let n be an odd natural number. Then, by (1.2), we get

$$\begin{aligned}&\int _{X}\chi \left( x\right) \left( 1+\lambda t\right) ^{\frac{nd+x}{\lambda }}d\mu _{-1}\left( x\right) +\int _{X}\chi \left( x\right) \left( 1+\lambda t\right) ^{\frac{x}{\lambda }}d\mu _{-1}\left( x\right) \\&\quad =2\sum _{l=0}^{nd-1}\left( -1\right) ^{l}\chi \left( l\right) \left( 1+\lambda t\right) ^{\frac{l}{\lambda }}. \nonumber \end{aligned}$$
(1.12)

Now, we set

$$\begin{aligned} R_{k}\left( n,\lambda \mid x\right) =2\sum _{l=0}^{n}\left( -1\right) ^{l}\chi \left( l\right) \left( l\mid \lambda \right) _{k}. \end{aligned}$$
(1.13)

From (1.2) and (1.12), we have

$$\begin{aligned}&\int _{X}\left( 1+\lambda t\right) ^{\frac{x+dn}{\lambda }}\chi \left( x\right) d\mu _{-1}\left( x\right) +\int _{X}\chi \left( x\right) \left( 1+\lambda t\right) ^{\frac{x}{\lambda }}d\mu _{-1}\left( x\right) \\&\quad =\frac{2\int _{X}\left( 1+\lambda t\right) ^{\frac{x}{\lambda }}\chi \left( x\right) d\mu _{-1}\left( x\right) }{\int _{X}\left( 1+\lambda t\right) ^{\frac{ndx}{\lambda }}d\mu _{-1}\left( x\right) }\nonumber \\&\quad =\sum _{k=0}^{\infty }R_{k}\left( nd-1,\lambda \mid \chi \right) \frac{t^{k}}{k!},\nonumber \end{aligned}$$
(1.14)

where \(n,d\in \mathbb {N}\) with \(n\equiv 1\pmod {2}\), \(d\equiv 1\pmod {2}\).

In this paper, we give some identities of symmetry for the generalized degenerate Euler polynomials attached to \(\chi \) derived from the symmetric properties of certain fermionic p-adic integrals on \(\mathbb {Z}_{p}\).

2 Identities of Symmetry for the Generalized Degenerate Euler Polynomials

Let \(w_{1},w_{2}\) be odd natural numbers. Then we consider the following integral equation:

$$\begin{aligned}&\frac{\int _{X}\int _{X}\left( 1+\lambda t\right) ^{\frac{w_{1}x_{1}+w_{2}x_{2}}{\lambda }}\chi \left( x_{1}\right) \chi \left( x_{2}\right) d\mu _{-1}\left( x_{1}\right) d\mu _{-1}\left( x_{2}\right) }{\int _{X}\left( 1+\lambda t\right) ^{\frac{dw_{1}w_{2}x}{\lambda }}d\mu _{-1}\left( x\right) }\\&\quad =\frac{2\left( \left( 1+\lambda t\right) ^{\frac{dw_{1}w_{2}}{\lambda }}+1\right) }{\left( \left( 1+\lambda t\right) ^{\frac{w_{1}d}{\lambda }}+1\right) \left( \left( 1+\lambda t\right) ^{\frac{w_{2}d}{\lambda }}+1\right) }\nonumber \\&\quad \mathrel { {{=}}}\times \sum _{a=0}^{d-1}\chi \left( a\right) \left( 1+\lambda t\right) ^{\frac{w_{1}a}{\lambda }}\left( -1\right) ^{a}\nonumber \\&\quad \mathrel { {{=}}}\times \sum _{b=0}^{d-1}\chi \left( b\right) \left( 1+\lambda t\right) ^{\frac{w_{2}b}{\lambda }}\left( -1\right) ^{b}.\nonumber \end{aligned}$$
(2.1)

From (1.10) and (1.11), we note that

$$\begin{aligned} \int _{X}\chi \left( y\right) \left( x+y\mid \lambda \right) _{n}d\mu _{-1}\left( y\right) =\mathcal {E}_{n,\lambda ,\chi }\left( x\right) ,\quad \left( n\ge 0\right) . \end{aligned}$$
(2.2)

By (1.14), we get

$$\begin{aligned} \int _{X}\chi \left( x\right) \left( x+dn\mid \lambda \right) _{k}d\mu _{-1}\left( x\right) +\int _{X}\chi \left( x\right) \left( x\mid \lambda \right) _{k}d\mu _{-1}\left( x\right) =R_{k}\left( nd-1,\lambda \mid x\right) , \end{aligned}$$
(2.3)

where \(k\ge 0\).

Thus, by (2.2) and (2.3), we get

$$\begin{aligned} \mathcal {E}_{k,\lambda ,\chi }\left( nd\right) +\mathcal {E}_{k,\lambda ,\chi }=R_{k}\left( nd-1,\lambda \mid \chi \right) , \end{aligned}$$
(2.4)

where \(k\ge 0\), \(n,d\in \mathbb {N}\) with \(n\equiv 1\pmod {2}\), \(d\equiv 1\pmod {2}\).

Now, we set

$$\begin{aligned} I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) =\frac{\int _{X}\int _{X}\chi \left( x_{1}\right) \chi \left( x_{2}\right) \left( 1+\lambda t\right) ^{\frac{w_{1}x_{1}+w_{2}x_{2}+w_{1}w_{2}x}{\lambda }}d\mu _{-1}\left( x_{1}\right) d\mu _{-1}\left( x_{2}\right) }{\int _{X}\left( 1+\lambda t\right) ^{\frac{dw_{1}w_{2}x}{\lambda }}d\mu _{-1}\left( x\right) }. \end{aligned}$$
(2.5)

From (2.5), we have

$$\begin{aligned}&I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \\&=\frac{2\left( \left( 1+\lambda t\right) ^{\frac{dw_{1}w_{2}}{\lambda }}+1\right) \left( 1+\lambda t\right) ^{\frac{w_{1}w_{2}x}{\lambda }}}{\left( \left( 1+\lambda t\right) ^{\frac{w_{1}d}{\lambda }}+1\right) \left( \left( 1+\lambda t\right) ^{\frac{w_{2}d}{\lambda }}+1\right) }\nonumber \\&\mathrel { {{=}}}\times \sum _{a=0}^{d-1}\chi \left( a\right) \left( -1\right) ^{a}\left( 1+\lambda t\right) ^{\frac{w_{1}a}{\lambda }}\nonumber \\&\mathrel { {{=}}}\times \sum _{b=0}^{d-1}\chi \left( b\right) \left( -1\right) ^{b}\left( 1+\lambda t\right) ^{\frac{w_{2}b}{\lambda }}.\nonumber \end{aligned}$$
(2.6)

Thus, by (2.6), we see that \(I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \) is symmetric in \(w_{1},w_{2}\). By (1.12), (1.14), (2.2) and (2.5), we get

$$\begin{aligned}&2I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \\&=\sum _{l=0}^{\infty }\left( \sum _{i=0}^{l}\left( {\begin{array}{c}l\\ i\end{array}}\right) \mathcal {E}_{i,\frac{\lambda }{w_{2}},\chi }\left( w_{1}x\right) w_{2}^{i}w_{1}^{l-i}R\left( \left. dw_{2}-1,\frac{\lambda }{w_{1}}\right| \chi \right) \right) \frac{t^{l}}{l!}.\nonumber \end{aligned}$$
(2.7)

From the symmetric property of \(I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \) in \(w_{1}\) and \(w_{2}\), we have

$$\begin{aligned}&2I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \\&\quad =2I_{\chi }\left( w_{2},w_{1}\mid \chi \right) \nonumber \\&\quad =\sum _{l=0}^{\infty }\left( \sum _{i=0}^{l}\left( {\begin{array}{c}l\\ i\end{array}}\right) \mathcal {E}_{i,\frac{\lambda }{w_{1}},\chi }\left( w_{2}x\right) w_{1}^{i}w_{2}^{l-i}R\left( \left. dw_{1}-1,\frac{\lambda }{w_{2}}\right| \chi \right) \right) \frac{t^{l}}{l!}.\nonumber \end{aligned}$$
(2.8)

Therefore, by (2.7) and (2.8), we obtain the following theorem.

Theorem 1

For \(w_{1},w_{2},d\in \mathbb {N}\) with \(w_{1}\equiv w_{2}\equiv d\equiv 1\pmod {2}\), let \(\chi \) be a Dirichlet character with conductor d. Then, we have

$$\begin{aligned}&\sum _{i=0}^{l}\left( {\begin{array}{c}l\\ i\end{array}}\right) \mathcal {E}_{i,\frac{\lambda }{w_{1}},\chi }\left( w_{2}x\right) w_{1}^{i}w_{2}^{l-i}R\left( \left. dw_{1}-1,\frac{\lambda }{w_{2}}\right| \chi \right) \\&\quad =\sum _{i=0}^{l}\left( {\begin{array}{c}l\\ i\end{array}}\right) \mathcal {E}_{i,\frac{\lambda }{w_{2}},\chi }\left( w_{1}x\right) w_{2}^{i}w_{1}^{l-i}R\left( \left. dw_{2}-1,\frac{\lambda }{w_{1}}\right| \chi \right) , \end{aligned}$$

where \(l\ge 0\).

When \(x=0\), by Theorem 1, we get

$$\begin{aligned}&\sum _{i=0}^{l}\left( {\begin{array}{c}l\\ i\end{array}}\right) \mathcal {E}_{i,\frac{\lambda }{w_{1}},\chi }w_{1}^{i}w_{2}^{l-i}R\left( \left. dw_{1}-1,\frac{\lambda }{w_{2}}\right| \chi \right) \\&\quad =\sum _{i=0}^{l}\left( {\begin{array}{c}l\\ i\end{array}}\right) \mathcal {E}_{i,\frac{\lambda }{w_{2}},\chi }w_{2}^{i}w_{1}^{l-i}R\left( \left. dw_{2}-1,\frac{\lambda }{w_{1}}\right| \chi \right) ,\quad \left( l\ge 0\right) . \end{aligned}$$

By (2.5), we get

$$\begin{aligned}&2I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \\&\quad =\sum _{l=0}^{dw_{2}-1}\left( -1\right) ^{l}\chi \left( l\right) \int _{X}\left( 1+\lambda t\right) ^{\frac{w_{2}}{\lambda }\left( w_{2}+w_{1}x+\frac{w_{1}}{w_{2}}l\right) }\chi \left( x_{2}\right) d\mu _{-1}\left( x\right) \nonumber \\&\quad =\sum _{n=0}^{\infty }\left( \sum _{l=0}^{dw_{2}-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\frac{\lambda }{w_{2}},\chi }\left( w_{1}x+\frac{w_{1}}{w_{2}}l\right) w_{2}^{n}\right) \frac{t^{n}}{n!}.\nonumber \end{aligned}$$
(2.9)

On the other hand,

$$\begin{aligned}&2I_{\chi }\left( w_{2},w_{1}\mid \lambda \right) =2I_{\chi }\left( w_{1},w_{2}\mid \lambda \right) \\&=\sum _{n=0}^{\infty }\left( \sum _{l=0}^{dw_{1}-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\frac{\lambda }{w_{1}},\chi }\left( w_{2}x+\frac{w_{2}}{w_{1}}l\right) w_{1}^{n}\right) \frac{t^{n}}{n!}.\nonumber \end{aligned}$$
(2.10)

Therefore, by (2.9) and (2.10), we obtain the following theorem.

Theorem 2

For \(w_{1},w_{2},d\in \mathbb {N}\) with \(d\equiv 1\pmod {2}\), \(w_{1}\equiv 1\pmod {2}\) and \(w_{2}\equiv 1\pmod {2}\), let \(\chi \) be a Dirichlet character with conductor d. Then, we have

$$\begin{aligned}&w_{2}^{n}\sum _{l=0}^{dw_{2}-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\frac{\lambda }{w_{2}},\chi }\left( w_{1}x+\frac{w_{1}}{w_{2}}l\right) \\&\quad =w_{1}^{n}\sum _{l=0}^{dw_{1}-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\frac{\lambda }{w_{1}},\chi }\left( w_{2}x+\frac{w_{2}}{w_{1}}l\right) ,\quad \left( n\ge 0\right) . \end{aligned}$$

To derive some interesting identities of symmetry for the generalized degenerate Euler polynomials attached to \(\chi \), we used the symmetric properties for certain fermionic p-adic integrals on \(\mathbb {Z}_{p}\). When \(w_{2}=1\), from Theorem 2, we have

$$\begin{aligned}&\sum _{l=0}^{d-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\lambda ,\chi }\left( w_{1}x+w_{1}l\right) \\&\quad =w_{1}^{n}\sum _{l=0}^{dw_{1}-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\frac{\lambda }{w_{1}},\chi }\left( x+\frac{1}{w_{1}}l\right) . \end{aligned}$$

In particular, for \(x=0\), we get

$$\begin{aligned}&\sum _{l=0}^{d-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\lambda ,\chi }\left( w_{1}l\right) \\&\quad =w_{1}^{n}\sum _{l=0}^{dw_{1}-1}\left( -1\right) ^{l}\chi \left( l\right) \mathcal {E}_{n,\frac{\lambda }{w_{1}},\chi }\left( \frac{1}{w_{1}}l\right) . \end{aligned}$$