Values of p-adic hypergeometric functions, and p-adic analogue of Kummer's linear identity

2023/01/25 by Neelam Saikia, Saikia, Neelam
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.10661

openalex publication_date 2023/01/25 · openalex created_date 2023/01/27 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime and \mathbbF_p be the finite field with p elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the p-adic setting which we denote by _3n-1G_3n-1(p, t), where n≥1 and t∈\mathbbF_p. These values are expressed in terms of numbers of zeros of certain polynomials over \mathbbF_p. These results lead to certain p-adic analogues of classical hypergeometric identities. Namely, we obtain p-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zeros of these functions. For instance, if n is odd then we obtain zeros of _3n-1G_3n-1(p, t)=0 under certain condition on t. In contrast we show that if n is even then the function _3n-1G_3n-1(p, t) has no non-trivial zeros for any prime p.

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