A modular supercongruence for _6F_5: An Apéry-like story

Abstract

Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION-PAS DE MODIFICATION 3.

Hypergeometric Supercongruences

“…2.1] for the case α = (1/2, 1/2, 1/2, 1/2, 1/2, 1/2). We learned at the conference that this example was recently studied further by Osburn, Straub, and Zudilin [7], who proved (4) below for s = 1 modulo p 3 and report that Mortenson conjectured it modulo p 5 .…”

On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series

“…Recently the arithmetic properties of truncated hypergeometric functions have been widely studied (cf. [7], [10]). In this paper, we shall consider the simplest truncated hypergeometric function…”

Interpolated Sequences and Critical L-Values of Modular Forms

“…It is not currently known if each of these cases has an interpolated version which is related (similar to (2)) to the critical L-value of a modular form of weight 4. Second, we echo the lament in [28] concerning the lack of algorithmic approaches in directly proving congruences, such as (15), between binomial sums. Third, can one extend the results in [19] to verify the cases in Example 3.7 and, more generally, find an explicit formula for the ratio L(f k , k − 1)/L(f k , 2) in terms of a rational number and a power of π?…”

“…Let D(x, k) be the summand in the sum defining A(x). Creative telescoping applied to D(x, k) determines the operator P (x, S x ) given in (28) as well as a rational function R(x, k) such that…”