, Volume 24, Issue 1, pp 93-128
Date: 25 Nov 2010

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

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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 4 F 3(1). We first show that \(K(\vec{x})\) is invariant under the action of a certain matrix group G K , isomorphic to the symmetric group S 6, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ7:e+f+gabcd=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 μ 1,μ 2,μ 3 of a certain matrix group M K , isomorphic to the Coxeter group W(D 6) (of order 23040) and containing the above group G K , 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 μ j ’s are in the same right coset of G K in M K . 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.

The set of \(\bigl({|M_{K}|/|G_{K}|\atop3}\bigr)=\bigl({32\atop3}\bigr)=4960\) resulting three-term relations may further be partitioned into five subsets, according to the Hamming type of the triple (μ 1,μ 2,μ 3) in question.

Each three-term relation of a given Hamming type may be transformed into any other of the same type by a change of variable. An explicit example of each of the five types of three-term relations is provided. It is seen that the number of monomials constituting the coefficient of a given K function, in a given three-term relation, increases geometrically with the Hamming distance between the other two K functions in that relation.

Two-term and three-term relations for a certain different linear combination of Saalschutzian 4 F 3(1) series have been studied elsewhere, by Whipple, Raynal, and others. Coxeter group theory allows us to highlight the structural differences between those situations and the present one, and generally provides for enhanced insight into the phenomenon of relations among hypergeometric series.

Further, previously established relations among terminating, Saalschutzian 4 F 3(1) series, and among not necessarily Saalschutzian 3 F 2(1) series, are seen to arise in straightforward ways as limits of the relations developed in this paper.