Hypergeometric series with gamma product formula

Abstract

Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this article we exploit the concepts of duality and reciprocity not only to extend already obtained results to a larger region but also to strengthen themselves substantially. Among other things we are able to settle the rationality and finiteness conjectures posed in the previous article.

Duality and Reciprocity for Hypergeometric Series With a Gamma Product Formula

“…Inspired by a series of works by Ebisu [2][3][4] we set out to develop a theoretical study of non-terminating Gauss hypergeometric series with one free parameter in [7]. It aims to find arithmetical constraints for a given hypergeometric sum to admit a gamma product formula (GPF).…”

“…This is a well-defined involution whenever r (r − p − q) does not vanish. As in our previous article [7], however, we restrict our attention to real data λ and think of duality and reciprocity as involutions on the real domain:…”

“…This restriction is due to the fact that in [7] we have developed a necessary asymptotic analysis of f (w; λ) only for real data λ, leaving the cases of complex data as a future task. The origin of these transformations is very simple.…”

“…In [7, §1] we introduced its subregions D = D − ∪ D 0 ∪ D + and E = E * − ∪ E * + ∪ E − * ∪ E + * , the figures of which are depicted in Figures 1 and 2 upon projected down to the (p, q)-plane with a fixed value of r > 0. Note that the study on D 0 is finished by assertion (1) of [7,Theorem 2.2]. We now introduce a new subregion F = F − ∪ F + with two components F − := { λ ∈ R 6 : p < 0, q < 0, r > 0; a, b ∈ R; 0 < x < 1 }, F + := { λ ∈ R 6 : p > r, q > r, r > 0; a, b ∈ R; 0 < x < 1 }.…”

“…, the figures of which are depicted in Figures 1 and 2 upon projection down to the ( p, q)-plane with a fixed value of r > 0. Note that the study on D 0 is finished by assertion (1) of [7,Theorem 2.2]. We now introduce a new subregion F = F − ∪ F + with two components:…”