Coxeter group actions on Saalschützian ₄F₃(1) series and very-well-poised ₇F₆(1) series

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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 ₄F₃(1) hypergeometric series or as a very-well-poised ₇F₆(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(D₅), which has 1920 elements, is an invariance group for L(x). The invariance relations for L(x) are classified into six types based on a double coset decomposition of the invariance group. The fundamental two-term relation is shown to generalize classical results about hypergeometric series. We derive Thomae's identity for ₃F₂(1) series, Bailey's identity for terminating Saalschützian ₄F₃(1) series, and Barnes' 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 23040 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.