Jackson's integral of multiple Hurwitz-Lerch zeta functions and multiple gamma functions

TL;DR

This paper develops a $q$-integral analogue of Raabe type formulas for multiple Hurwitz-Lerch zeta and gamma functions using Jackson integrals, extending previous results and revealing new relationships among zeta functions.

Contribution

It introduces a novel $q$-integral framework for multiple zeta and gamma functions, generalizing existing formulas and establishing new interrelations.

Findings

Derived $q$-analogues of Raabe formulas for multiple zeta and gamma functions.

Extended previous single-variable results to multiple-variable functions.

Discovered new relationships between higher and lower order Hurwitz zeta functions.

Abstract

Using the Jackson integral, we obtain the $q$-integral analogue of the Raabe type formulas for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize $q$-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [N. Kurokawa, K. Mimachi, and M. Wakayama, Jackson's integral of the Hurwitz zeta function, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 43--56]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.