Coxeter group actions on Saalschützian series and very-well-poised series

In this paper we consider a function , which can be written as a linear combination of two Saalschützian hypergeometric series or as a very-well-poised 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(D5), which has 1920 elements, is an invariance group for . The invariance relations for 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 series, Baileyʼs identity for terminating Saalschützian 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(D6), which has 23 040 elements. Based on the right cosets of W(D5) in W(D6), 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.