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

Abstract

Suppose that $ω_1,\ldots,ω_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ with parameter vector $\barω=(ω_1,\ldots,ω_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ ζ_{E,N}(s,x;\barω)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+ω_1t_1+\cdots+ω_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ which we denote by $ζ_{p,E,N}(s,x;\barω).$ We prove several properties of $ζ_{p,E,N}(s,x; \barω)$, 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 $E_{N,n}(x;\barω)$ $p$-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 ${\rm Log}\, Γ_{\! D,E,N}(x;\barω)$ 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.

27 pages