Zeros of $p$-adic hypergeometric functions, $p$-adic analogues of Kummer's and Pfaff's identities

Neelam Saikia

Vol. 307 (2020), No. 2, 491–510

DOI: 10.2140/pjm.2020.307.491

Abstract

We classify all the zeros and nonzero values of a family of hypergeometric series in the $p$-adic setting. These values of hypergeometric series in the $p$-adic setting lead to transformations of hypergeometric series in the $p$-adic setting which can be described as $p$-adic analogues of Kummer’s and Pfaff’s linear transformations on classical hypergeometric series. We also evaluate certain summation identities for hypergeometric series in the $p$-adic setting as well as Gaussian hypergeometric series.

Keywords

character sum, Gauss sums, Jacobi sums, $p$-adic gamma function

Mathematical Subject Classification 2010

Primary: 11S80, 11T24, 33C20, 33C99, 33E50

Milestones

Received: 19 March 2020

Revised: 1 May 2020

Accepted: 1 May 2020

Published: 4 September 2020