Proof of some supercongruences concerning truncated hypergeometric series

TL;DR

This paper proves new supercongruences involving truncated hypergeometric series, confirming conjectures by Sun and Guo, and advances understanding of hypergeometric series modulo prime powers.

Contribution

It establishes two specific supercongruences for truncated hypergeometric series that confirm prior conjectures, expanding the theory of hypergeometric supercongruences.

Findings

Proved a supercongruence for a hypergeometric series involving (1/2)_k^3.

Confirmed a second supercongruence involving (1/2)_k^4.

Validated conjectures by Sun and Guo regarding hypergeometric supercongruences.

Abstract

In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime $p>3$ and positive integer $r$, $$ \sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac12)_k^3}{(1)_k^3}4^k\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}} $$ and $$ \sum_{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac12)_k^4}{(1)_k^4}\equiv p^r+\frac76p^{r+3}B_{p-3}\pmod{p^{r+4}}, $$ where $(x)_k=x(x+1)\cdots(x+k-1)$ is the Pochhammer symbol and $B_0,B_1,B_2,\ldots$ are Bernoulli numbers. These two congruences confirm conjectures of Sun [Sci. China Math. 54 (2011), 2509--2535] and Guo [Adv. Appl. Math. 120 (2020), Art. 102078], respectively.