Generalizations of two hypergeometric sums related to conjectures of Guo

TL;DR

This work generalizes Guo's hypergeometric-sum conjectures by deriving two finite-sum identities parameterized by integers $\\ell\\ge1$, $s\\ge0$, and $M\\ge s$, using the WZ-method and Zeilberger's algorithm. The first identity is established via a WZ-pair and yields a boundary expression with rising factorials $(1+\\frac{1}{\\ell})_{M+\\ast}$, while the second identity is obtained through Zeilberger's algorithm, giving a squared boundary term. Together, these results broaden the toolkit for proving related supercongruences and hypergeometric-sum generalizations tied to Guo's conjectures, and they connect to the $p$-adic analogues studied in the literature. The methods provide structured, algorithmic paths to closed-form evaluations of intricate hypergeometric sums.

Abstract

In 2021, the first author and Kalita obtained two general hypergeometric formulas for sums involving certain rising factorials to prove some supercongruence conjectures of Guo related to (B.2) and (C.2). In this paper, we further generalize those formulas by using the WZ-method and the Zeilberger algorithm, respectively.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof : Proof of theorem \ref{['thmguo1a']}
• Lemma 3.1
• proof
• proof : Proof of theorem \ref{['thmguo1b']}