Bivariate Identities for Values of the Hurwitz Zeta Function and Supercongruences

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.

New series for powers of <inline-formula><tex-math id="M1">$ \pi $</tex-math></inline-formula> and related congruences

“…The second, third and fourth identities were obtained by J. Guillera [17] in 2008. The fifth identity on K was conjectured by Sun [33] and later confirmed by K. Hessami Pilehrood and T. Hessami Pilehrood [22] in 2012. The last four identities were also conjectured by Sun [33], and they were later proved in the paper […”