On the Proof of the Genčev-Rucki Conjecture for Multiple Apéry-Like Series

Abstract

In this paper, we employ the theories and techniques of hypergeometric functions to provide two distinct proofs of the conjectured identities involving multiple Apéry-like series with central binomial coefficients and multiple harmonic star sums, as recently proposed by Genčev and Rucki. Furthermore, we establish several more general identities for multiple Apéry-like series. Furthermore, by utilizing the method of iterated integrals, a class of multiple mixed values can be expressed as combinations of the multiple Apéry-like series identities conjectured by Genčev and Rucki and $\zeta (2,\dots ,2)$, thus allowing explicit formulas for these multiple mixed values to be derived in terms of Riemann zeta values.