Summation identities and transformations for hypergeometric series

2016/09/22 by Rupam Barman, Barman, Rupam, Neelam Saikia +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1609.06829

arxiv created 2016/09/22 · arxiv updated 2016/09/23

Abstract

We find summation identities and transformations for the McCarthy's p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family Z_λ: x₁^d+x₂^d=dλx₁x₂^d-1 over a finite field \mathbbF_p. A. Salerno expresses the number of points over a finite field \mathbbF_p on the family Z_λ in terms of quotients of p-adic gamma function under the condition that d|p-1. In this paper, we first express the number of points over a finite field \mathbbF_p on the family Z_λ in terms of McCarthy's p-adic hypergeometric series for any odd prime p not dividing d(d-1), and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene's finite field hypergeometric series.

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