2020/12/21 by Neelam Saikia, Saikia, Neelam
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2012.11173
openalex publication_date 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be an odd prime and \mathbbFp be the finite field with p elements. McCarthy \citemccarthy-pacific initiated a study of hypergeometric functions in the p-adic setting. This function can be understood as p-adic analogue of Gauss' hypergeometric function, and also some kind of extension of Greene's hypergeometric function over \mathbbFp. In this paper we investigate values of two generic families of McCarthy's hypergeometric functions denoted by nGn(t), and n\widetildeGn(t) for n≥3, and t∈\mathbbFp. The values of the function nGn(t) certainly depend on whether t is n-th power residue modulo p or not. Similarly, the values of the function n\widetildeGn(t) rely on the incongruent modulo p solutions of yn-yn-1+\frac(n-1)n-1tnn≡0\pmodp. These results generalize special cases of p-adic analogues of Whipple's theorem and Dixon's theorem of classical hypergeometric series. We examine zeros of the functions nGn(t), and n\widetildeGn(t) over \mathbbFp. Moreover, we look into the values of t for which nGn(t)=0 for infinitely many primes. For example, we show that there are infinitely many primes for which 2kG2k(-1)=0. In contrast, for t≠0 there is no prime for which 2k\widetildeG2k(t)=0.