On $p$-adic multiple Barnes-Euler zeta functions and the corresponding log gamma functions

TL;DR

This paper introduces a new $p$-adic multiple Barnes-Euler zeta function, explores its properties, and connects it to higher order Euler polynomials and a $p$-adic Log Gamma function, expanding the theory of $p$-adic special functions.

Contribution

It defines the $p$-adic analogue of the multiple Barnes-Euler zeta function and establishes its key properties, including functional equations and interpolation of Euler polynomials.

Findings

The $p$-adic zeta function admits a Laurent series expansion.

It interpolates higher order Euler polynomials at nonpositive integers.

The associated $p$-adic Log Gamma function has integral representations and satisfies key functional equations.

Abstract

Suppose that $\omega_1,\ldots,\omega_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $\zeta_{E,N}(s,x;\bar\omega)$ with parameter vector $\bar\omega=(\omega_1,\ldots,\omega_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ \zeta_{E,N}(s,x;\bar\omega)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+\omega_1t_1+\cdots+\omega_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $\zeta_{E,N}(s,x;\bar\omega)$ which we denote by $\zeta_{p,E,N}(s,x;\bar\omega).$ We prove several properties of $\zeta_{p,E,N}(s,x; \bar\omega)$, including the convergent Laurent series expansion, the distribution formula, the difference equation, the reflection functional equation and the derivative formula. By computing the values of this kind of $p$-adic zeta function at nonpositive integers, we show that it interpolates the higher order Euler polynomials $E_{N,n}(x;\bar\omega)$ $p$-adically. Furthermore, we define the corresponding multiple $p$-adic Diamond-Euler Log Gamma function. We also show that the multiple $p$-adic Diamond-Euler Log Gamma function ${\rm Log}\, \Gamma_{\! D,E,N}(x;\bar\omega)$ has an integral representation by the multiple fermionic $p$-adic integral, and it satisfies the distribution formula, the difference equation, the reflection functional equation, the derivative formula and also the Stirling's series expansions.