The Ramanujan Journal

, Volume 24, Issue 1, pp 93–128

Coxeter group actions on 4F3(1) hypergeometric series

Article

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 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.

Keywords

Hypergeometric series Coxeter groups 

Mathematics Subject Classification (2000)

33C20 33C60 33C80 

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References

  1. 1.
    Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935) MATHGoogle Scholar
  2. 2.
    Barnes, E.W.: A new development of the theory of hypergeometric functions. Proc. Lond. Math. Soc. 2(6), 141–177 (1908) CrossRefGoogle Scholar
  3. 3.
    Barnes, E.W.: A transformation of generalized hypergeometric series. Q. J. Math. 41, 136–140 (1910) MATHGoogle Scholar
  4. 4.
    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) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bump, D.: Barnes’ second lemma and its application to Rankin–Selberg convolutions. Am. J. Math. 109, 179–186 (1987) MathSciNetGoogle Scholar
  6. 6.
    Drake, G. (ed.): Springer Handbook of Atomic, Molecular, and Optical Physics. Springer, New York (2006) Google Scholar
  7. 7.
    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) Google Scholar
  8. 8.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Academic Press, New York (2007) Google Scholar
  9. 9.
    Grozin, A.: Lectures on QED and QCD: Practical Calculation and Renormalization of One- and Multi-Loop Feynman Diagrams. Singapore, World Scientific (2007) Google Scholar
  10. 10.
    Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990) MATHGoogle Scholar
  11. 11.
    Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Englewood Cliffs, Prentice-Hall (1970) MATHGoogle Scholar
  12. 12.
    Lievens, S., Van der Jeugt, J.: Invariance groups of three term transformations for basic hypergeometric series. J. Comput. Appl. Math. 197, 1–14 (2006) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    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) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Raynal, J.: On the definition and properties of generalized 6-j symbols. J. Math. Phys. 20(12), 2398–2415 (1979) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Stade, E.: Hypergeometric series and Euler factors at infinity for L-functions on GL(3,ℝ)×GL(3,ℝ). Am. J. Math. 115(2), 371–387 (1993) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Stade, E.: Mellin transforms of Whittaker functions on GL(4,ℝ) and GL(4,ℂ). Manuscr. Math. 87, 511–526 (1995) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Stade, E.: Mellin transforms of GL(n,ℝ) Whittaker functions. Am. J. Math. 123, 121–161 (2001) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Stade, E.: Archimedean L-factors on GL(n)×GL(n) and generalized Barnes integrals. Isr. J. Math. 127, 201–220 (2002) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Stade, E., Taggart, J.: Hypergeometric series, a Barnes-type lemma, and Whittaker functions. J. Lond. Math. Soc. 61, 133–152 (2000) CrossRefMathSciNetGoogle Scholar
  20. 20.
    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) Google Scholar
  21. 21.
    Van de Bult, F.J., Rains, E.M., Stokman, J.V.: Properties of generalized univariate hypergeometric functions. Commun. Math. Phys. 275, 37–95 (2007) MATHCrossRefGoogle Scholar
  22. 22.
    Van der Jeugt, J., Rao, K.S.: Invariance groups of transformations of basic hypergeometric series. J. Math. Phys. 40, 6692–6700 (1999) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    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) Google Scholar
  24. 24.
    Whipple, F.J.W.: Relations between well-poised hypergeometric series of the type 7F6. Proc. Lond. Math. Soc. 40(2), 336–344 (1936) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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