Coxeter group actions on Saalschützian F34(1) series and very-well-poised F67(1) series

摘要

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 F34(1) hypergeometric series or as a very-well-poised F67(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 Thomae's identity for F23(1) series, Bailey's identity for terminating Saalschützian F34(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(D_6), which has 23 040 elements. Based on the right cosets of W(D5) 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.