Coxeter group actions on 4 F 3(1) hypergeometric series

Abstract

In this paper we investigate a certain linear combination \(K(\vec{x})=K(a;b,c,d;e,f,g)\) of two Saalschutzian hypergeometric series of type 4F3(1). We first show that \(K(\vec{x})\) is invariant under the action of a certain matrix group GK, isomorphic to the symmetric group S6, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ7:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ1,μ2,μ3 of a certain matrix group MK, isomorphic to the Coxeter group W(D6) (of order 23040) and containing the above group GK, there is a relation among \(K(\mu_{1}\vec{x})\), \(K(\mu_{2}\vec{x})\), and \(K(\mu_{3}\vec{x})\), provided that no two of the μj’s are in the same right coset of GK in MK. The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in a,b,c,d,e,f,g.