Coxeter group actions on _{4} F _{3}(1) hypergeometric series
 Marc Formichella,
 R. M. Green,
 Eric Stade
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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)∈ℂ^{7}:e+f+g−a−b−c−d=1}. We further develop an algebra of threeterm 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 threeterm 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 threeterm 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 threeterm 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 threeterm relations is provided. It is seen that the number of monomials constituting the coefficient of a given K function, in a given threeterm relation, increases geometrically with the Hamming distance between the other two K functions in that relation.
Twoterm and threeterm 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.
 Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)
 Barnes, E.W.: A new development of the theory of hypergeometric functions. Proc. Lond. Math. Soc. 2(6), 141–177 (1908) CrossRef
 Barnes, E.W.: A transformation of generalized hypergeometric series. Q. J. Math. 41, 136–140 (1910)
 Beyer, W.A., Louck, J.D., Stein, P.R.: Group theoretical basis of some identities for the generalized hypergeometric series. J. Math. Phys. 28(3), 497–508 (1987) CrossRef
 Bump, D.: Barnes’ second lemma and its application to Rankin–Selberg convolutions. Am. J. Math. 109, 179–186 (1987)
 Drake, G. (ed.): Springer Handbook of Atomic, Molecular, and Optical Physics. Springer, New York (2006)
 Gauss, C.F.: Disquisitiones generales circa seriem infinitam \(1+\frac{\alpha\beta }{1\cdot\gamma}x+\frac{\alpha(\alpha+1)\beta (\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}x x+ \mbox{etc}\) . In: Werke, vol. 3, pp. 123–162. Königliche Gesellschaft der Wissenschaften, Göttingen (1876)
 Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Academic Press, New York (2007)
 Grozin, A.: Lectures on QED and QCD: Practical Calculation and Renormalization of One and MultiLoop Feynman Diagrams. Singapore, World Scientific (2007)
 Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
 Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Englewood Cliffs, PrenticeHall (1970)
 Lievens, S., Van der Jeugt, J.: Invariance groups of three term transformations for basic hypergeometric series. J. Comput. Appl. Math. 197, 1–14 (2006) CrossRef
 Lievens, S., Van der Jeugt, J.: Symmetry groups of Bailey’s transformations for _{10} φ _{9}series. J. Comput. Appl. Math. 206, 498–519 (2007) CrossRef
 Raynal, J.: On the definition and properties of generalized 6j symbols. J. Math. Phys. 20(12), 2398–2415 (1979) CrossRef
 Stade, E.: Hypergeometric series and Euler factors at infinity for Lfunctions on GL(3,ℝ)×GL(3,ℝ). Am. J. Math. 115(2), 371–387 (1993) CrossRef
 Stade, E.: Mellin transforms of Whittaker functions on GL(4,ℝ) and GL(4,ℂ). Manuscr. Math. 87, 511–526 (1995) CrossRef
 Stade, E.: Mellin transforms of GL(n,ℝ) Whittaker functions. Am. J. Math. 123, 121–161 (2001) CrossRef
 Stade, E.: Archimedean Lfactors on GL(n)×GL(n) and generalized Barnes integrals. Isr. J. Math. 127, 201–220 (2002) CrossRef
 Stade, E., Taggart, J.: Hypergeometric series, a Barnestype lemma, and Whittaker functions. J. Lond. Math. Soc. 61, 133–152 (2000) CrossRef
 Thomae, J.: Ueber die funktionen welche durch Reihen der form dargestellt werden: \(1+\frac{p p' p }{1 q' q }+\cdots\) . J. Math. 87, 26–73 (1879)
 Van de Bult, F.J., Rains, E.M., Stokman, J.V.: Properties of generalized univariate hypergeometric functions. Commun. Math. Phys. 275, 37–95 (2007) CrossRef
 Van der Jeugt, J., Rao, K.S.: Invariance groups of transformations of basic hypergeometric series. J. Math. Phys. 40, 6692–6700 (1999) CrossRef
 Whipple, F.J.W.: A group of generalized hypergeometric series; relations between 120 allied series of the type F(a,b,c;e,f). Proc. Lond. Math. Soc. 23(2), 247–263 (1925)
 Whipple, F.J.W.: Relations between wellpoised hypergeometric series of the type 7F6. Proc. Lond. Math. Soc. 40(2), 336–344 (1936) CrossRef
 Title
 Coxeter group actions on _{4} F _{3}(1) hypergeometric series
 Journal

The Ramanujan Journal
Volume 24, Issue 1 , pp 93128
 Cover Date
 20110101
 DOI
 10.1007/s1113901092532
 Print ISSN
 13824090
 Online ISSN
 15729303
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Hypergeometric series
 Coxeter groups
 33C20
 33C60
 33C80
 Authors

 Marc Formichella ^{(1)}
 R. M. Green ^{(1)}
 Eric Stade ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Colorado, Boulder, CO, 80309, USA