Apéry numbers, Jacobi sums, and special values of generalized \(p\)-adic hypergeometric functions
type: Item
instance of: scholarly article
title: Apéry numbers, Jacobi sums, and special values of generalized \(p\)-adic hypergeometric functions
zbMATH Open document ID: 0759.11042
DOI: 10.1016/0022-314X(92)90123-7
author: Paul Thomas Young
published in: Journal of Number Theory
publication date: 1993-01-16T00:00:00Z
review text: Let \(\mathbb{Z}_ p\) be the set of \(p\)-adic integers and define the map \(a\mapsto a'\) on \(\mathbb{Q}\cap\mathbb{Z}_ p\) by requiring that \(pa'- a\in\{0,1,\dots,p-1\}\). Let \(_ kF_{k-1}\) be the hypergeometric function in one variable with parameters \(a_ 1,\dots,a_ k\); \(c_ 1,\dots,c_{k-1}\), where the \(c_ i\) are neither zero nor a negative integer. Suppose also that the \(c_ i\) are \(p\)-adic units. Suppose also that \(_ kF_{k-1}\), as a \(p\)-adic function, is bounded on the unit disc. Then a theorem of Dwork states that \(_ kF_{k-1}/_ k\tilde F_{k-1}\) can be continued analytically to the closed unit disc. Here \(_ k\tilde F_{k-1}\) denotes the function obtained by replacing the \(a_ j\), \(c_ i\) by \(a_ j'\), \(c_ i'\). Denote this continuation by \(_ k{\mathcal F}_{k-1}\). In this paper the author generalises several classical evaluations of special values of \(_ kF_{k-1}\) to \(p\)-adic evaluations of \(_ kF_{k-1}/_ k\tilde F_{k-1}\). Examples are Kummer's evaluation of certain \(_ 2F_ 1\) at \(-1\) and the values at 1 of Saalschützian and well-posed \(_ 3F_ 2\). The author also notes that certain congruences obtained from formal group theory applied to families of varieties (Atkin, Swinnerton-Dyer congruences) can be reformulated elegantly using values of \(_ k{\mathcal F}_{k-1}\). This yields some unexpected further evaluations of \(_ k{\mathcal F}_{k-1}\).
reviewed by: Frits Beukers
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 12H25
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 89654
zbMATH Keywords: \(p\)-adic gamma function
zbMATH Keywords: zeta-function
zbMATH Keywords: Jacobi sums
zbMATH Keywords: Apéry numbers
zbMATH Keywords: unit roots of Hecke polynomials
zbMATH Keywords: \(p\)-adic evaluations
zbMATH Keywords: special values
zbMATH Keywords: congruences
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 89654