#### Vol. 307, No. 2, 2020

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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
##### 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
##### Authors
 Neelam Saikia Department of Mathematics Indian Institute of Technology Guwahati North Guwahati Assam India