On the Proof of the Genčev-Rucki Conjecture for Multiple Apéry-Like Series
Ce Xu
TL;DR
The paper addresses the Genčev-Rucki conjecture on sums of multiple Apéry-like series with central binomial coefficients and multiple harmonic star sums, notably the identity $\sum_{n=1}^\infty \frac{\binom{2n}{n}}{n4^n}\zeta_n^\star(\{2\}_r)=2(1-4^{-r})\zeta(2r+1)$. It develops two distinct hypergeometric-function proofs to establish this conjecture and explores generalized identities for related Apéry-like series. Additionally, the authors use iterated-integral methods to express a class of multiple mixed values as combinations of these identities and $\zeta$-values, yielding explicit formulas in terms of $\zeta$-values and polylogarithms. A complementary proof employing partial fraction decomposition provides parametric evaluations of ${}_3F_2\left({1,1,3/2\atop 2-x,2+x};1\right)$, including digamma representations, and confirms consistency with the hypergeometric approach. Overall, the work strengthens the connections between Apéry-like series, hypergeometric transformations, iterated integrals, and zeta-values, offering new exact results and techniques for studying multiple zeta phenomena.
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 $ζ(2,\ldots,2)$, thus allowing explicit formulas for these multiple mixed values to be derived in terms of Riemann zeta values.