Supercongruences for sporadic sequences

Abstract

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

Divisibility properties of sporadic Apéry-like numbers

“…Remarkably, these sequences are again connected to modular forms [12] (the subscript refers to the level) and satisfy supercongruences, which are proved in [23]. Indeed, it was the corresponding modular forms and Ramanujan-type series for 1/π that led Cooper to study these sequences, and the binomial expressions for s 7 and s 18 were found subsequently by Zudilin (sequence s 10 was well-known before).…”

“…In his surprising proof [5,24] where n = n 0 + n 1 p + · · · + n r p r is the expansion of n in base p. Initial work of F. Beukers [8] and D. Zagier [29], which was extended by G. Almkvist, W. Zudilin [4] and S. Cooper [12], has complemented the Apéry numbers with a, conjecturally finite, set of sequences, known as Apéry-like, which share (or are believed to share) many of the remarkable properties of the Apéry numbers, such as connections to modular forms [2,7,27] or supercongruences [6,10,13,[21][22][23]. After briefly reviewing Apéry-like sequences in Section 2, we prove in Sections 3 and 4 our main result that all of these sequences also satisfy the Lucas congruences (1.2).…”

Some supercongruences on truncated hypergeometric series

“…Some similar types of supercongruences on combinatorial numbers such as Almkvist-Zudilin numbers, Domb numbers and Apéry-like numbers have been studied by several authors, see for example, Amdeberhan and Tauraso [2], Chan, Cooper and Sica [5], Osburn and Sahu [16], and Osburn, Sahu and Straub [17]. In this paper, we aim to establish the same type of supercongruences on truncated 3 F 2 hypergeometric series.…”

Congruences for generalized Apéry numbers and Gaussian hypergeometric series

“…In Section 2, we prove Theorem 1.1 using the techniques of [13] and [24]. We prove Theorem 1.2 in Section 3, and deduce the congruence relations for A( …”

“…Thus we are left to prove the case when l = 2. We now follow the approach of [24]. For l = 2, we need to show that…”