Coxeter group actions on Saalschützian [formula omitted] series and very-well-poised [formula omitted] series

Abstract

In this paper we consider a function L ( x → ) = L ( a , b , c , d ; e ; f , g ) , which can be written as a linear combination of two Saalschützian F 3 4 ( 1 ) hypergeometric series or as a very-well-poised F 6 7 ( 1 ) hypergeometric series. We explore two-term and three-term relations satisfied by the L function and put them in the framework of group theory. We prove a fundamental two-term relation satisfied by the L function and show that this relation implies that the Coxeter group W ( D 5 ) , which has 1920 elements, is an invariance group for L ( x → ) . The invariance relations for L ( x → ) are given two classifications based on two double coset decompositions of the invariance group. The fundamental two-term relation is shown to generalize classical results about hypergeometric series. We derive Thomaeʼs identity for F 2 3 ( 1 ) series, Baileyʼs identity for terminating Saalschützian F 3 4 ( 1 ) series, and Barnesʼ second lemma as consequences. We further explore three-term relations satisfied by L ( a , b , c , d ; e ; f , g ) . The group that governs the three-term relations is shown to be isomorphic to the Coxeter group W ( D 6 ) , which has 23 040 elements. Based on the right cosets of W ( D 5 ) in W ( D 6 ) , we demonstrate the existence of 220 three-term relations satisfied by the L function that fall into two families according to the notion of L-coherence. The complexity of the coefficients in front of the L functions in the three-term relations is studied and is shown to also depend on L-coherence.