On $p$-adic Hurwitz-type Euler zeta functions

TL;DR

This paper introduces a new class of $p$-adic Hurwitz-type Euler zeta functions, explores their properties, and connects them to $p$-adic Euler $L$-functions, enriching the theory of $p$-adic special functions.

Contribution

It defines $p$-adic Hurwitz-type Euler zeta functions using fermionic $p$-adic integrals and establishes their key properties and connections to $p$-adic Euler $L$-functions.

Findings

Interpolates Euler polynomials $p$-adically at negative integers

Provides Laurent series, functional, reflection, and derivative formulas

Defines $p$-adic Euler $L$-functions via these zeta functions

Abstract

The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$-functions has also been given by using the $p$-adic Hurwitz-type Euler zeta functions.