Supercongruences concerning truncated hypergeometric series

Abstract

In this paper, we mainly establish two supercongruences involving truncated hypergeometric series by using some hypergeometric transformation formulas. The first supercongruence confirms a recent conjecture of the second author. The second supercongruence confirms a conjecture of Guo, Liu and Schlosser partially, and gives a parametric extension of a supercongruence of Long and Ramakrishna.

Factors of certain basic hypergeometric sums

“…where Γ p (x) denotes the p-adic Gamma function. Wang and Pan [13] further proved that, for d ⩾ 3, the supercongruence (1.4) also holds modulo p 3 , confirming a conjecture in [2]. A q-analogue of (1.4) was given by the second author [5].…”

Some congruences from the Karlsson-Minton summation formula

“…In fact, Deines et al also conjectured that for any integer d ≥ 3 and prime p ≡ 1 (mod d), (1.3) holds modulo p 3 , and this conjecture was later confirmed by the first author and Pan [14].…”