The Ramanujan Journal

, Volume 24, Issue 1, pp 93–128

Coxeter group actions on 4F3(1) hypergeometric series



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 4F3(1). We first show that \(K(\vec{x})\) is invariant under the action of a certain matrix group GK, isomorphic to the symmetric group S6, 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 μ1,μ2,μ3 of a certain matrix group MK, isomorphic to the Coxeter group W(D6) (of order 23040) and containing the above group GK, 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 μj’s are in the same right coset of GK in MK. 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 (μ1,μ2,μ3) 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 4F3(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 4F3(1) series, and among not necessarily Saalschutzian 3F2(1) series, are seen to arise in straightforward ways as limits of the relations developed in this paper.


Hypergeometric series Coxeter groups 

Mathematics Subject Classification (2000)

33C20 33C60 33C80 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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