Apéry numbers, Jacobi sums, and special values of generalized \(p\)-adic hypergeometric functions
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 \(_ k{\mathcal F}_{k-1}\).
scientific article; zbMATH DE number 89654
scientific article