Some special values of hypergeometric series related to central values of automorphic $L$-functions

TL;DR

This paper investigates special hypergeometric series values linked to the positivity of central automorphic L-function values, providing partial answers through orthogonal polynomial theory and hypergeometric relations.

Contribution

It introduces a novel approach connecting hypergeometric series with automorphic L-function positivity using orthogonal polynomials and three-term hypergeometric relations.

Findings

Partial answers to positivity questions of automorphic L-values

Application of orthogonal polynomial theory to hypergeometric series

Development of new relations for hypergeometric series

Abstract

In a recent work, H.Narita presented problems concerning the strict positivity of central values of certain automorphic $L$-functions in the form of questions regarding special values of the hypergeometric series. In this paper, we present partial answers to these questions using the theory of orthogonal polynomials and three term relations of the hypergeometric series.