Bivariate identities for values of the Hurwitz zeta function and supercongruences

TL;DR

This paper introduces a new identity for Hurwitz zeta function values, extending known identities and applying Markov-WZ pairs to prove supercongruence conjectures by Guillera, Zudilin, and Sun.

Contribution

It presents a novel identity for Hurwitz zeta values, extending Cohen's bivariate identity, and applies this to prove several supercongruence conjectures.

Findings

New identity for Hurwitz zeta function values

Extension of Cohen's bivariate identity

Proof of supercongruence conjectures

Abstract

In this paper, we prove a new identity for values of the Hurwitz zeta function which contains as particular cases Koecher's identity for odd zeta values, the Bailey-Borwein-Bradley identity for even zeta values and many other interesting formulas related to values of the Hurwitz zeta function. We also get an extension of the bivariate identity of Cohen to values of the Hurwitz zeta function. The main tool we use here is a construction of new Markov-WZ pairs. As application of our results, we prove several conjectures on supercongruences proposed by J. Guillera, W. Zudilin, and Z.-W. Sun.