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

2025/10/10 by Xu, Ce
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.09052

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,…,2), thus allowing explicit formulas for these multiple mixed values to be derived in terms of Riemann zeta values.

Citations

• Generating functions of multiple t-star values of general level
• Elementary proofs of Zagier's formula for multiple zeta values and its odd variant
• Explicit Relations of Some Variants of Convoluted Multiple Zeta Values
• Another look at Zagier's formula for multiple zeta values involving Hoffman elements
• On the interplay between hypergeometric series, Fourier-Legendre\n expansions and Euler sums

Related

• Parametric Apéry-type Series and Hurwitz-type Multiple Zeta Values
• Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series
• Multiple zeta values and multiple Apéry-like sums
• Explicit Relations between Multiple Zeta Values and Related Variants
• Apéry-Like Sums and Colored Multiple Zeta Values