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

Abstract

Suppose that ${\omega }_1,\dots ,{\omega }_N$ 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 $\bar{\omega }=({\omega }_1,\dots ,{\omega }_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $${\zeta }_{E,N}(s,x;\bar{\omega })=\sum _{t_1=0}^{\infty }\cdots \sum _{t_N=0}^{\infty }\frac{(-1{)}^{t_1+\cdots +t_N}}{(x+{\omega }_1{t}_1+\cdots +{\omega }_N{t}_N{)}^s}.$$ 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 $\mathrm{Log}{\Gamma }_{D,E,N}(x;\bar{\omega })$ 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.

27 pages

References in corpus (1)

• Barnes type multiple q-zeta functions and q-Euler polynomials

Cited by in corpus (1)

• Generalizations of Lerch's formula by Barnes' multiple zeta functions