On the Proof of the Gen\v{c}ev-Rucki Conjecture for Multiple Ap\'ery-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 Ap\'ery-like series with central binomial coefficients and multiple harmonic star sums, as recently proposed by Gen\v{c}ev and Rucki. Furthermore, we establish several more general identities for multiple Ap\'ery-like series. Furthermore, by utilizing the method of iterated integrals, a class of multiple mixed values can be expressed as combinations of the multiple Ap\'ery-like series identities conjectured by Gen\v{c}ev and Rucki and $\zeta(2,\ldots,2)$, thus allowing explicit formulas for these multiple mixed values to be derived in terms of Riemann zeta values.