Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta distributions

TL;DR

This paper introduces Hurwitz-Lerch zeta distributions using the Gamma function and extends the Hurwitz-Lerch type of Euler-Zagier double zeta distributions beyond the region of absolute convergence.

Contribution

It defines new zeta distributions based on Hurwitz-Lerch and Euler-Zagier functions, expanding their applicability beyond traditional convergence regions.

Findings

Defined Hurwitz-Lerch zeta distributions for 0 < σ ≠ 1.

Extended Euler-Zagier double zeta distributions outside absolute convergence.

Provided mathematical framework for these distributions.

Abstract

In this paper, we give Hurwitz-Lerch zeta distributions with $0 < \sigma \ne 1$ by using the Gamma function. Moreover, we define Hurwitz-Lerch type of Euler-Zagier double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence.