Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 039, 17 pages      arXiv:1204.0254

Some Remarks on Very-Well-Poised ${}_8\phi_7$ Series

Jasper V. Stokman
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

Received April 05, 2012, in final form June 18, 2012; Published online June 27, 2012

Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised ${}_8\phi_7$ series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new derivation of a known quadratic transformation formula for very-well-poised ${}_8\phi_7$ series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions.

Key words: very-well-poised basic hypergeometric series; Askey-Wilson functions; quadratic transformation formulas; theta functions.

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  1. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319.
  2. van de Bult F.J., Ruijsenaars' hypergeometric function and the modular double of ${\mathcal U}_q(\mathfrak{sl}_2({\mathbb C}))$, Adv. Math. 204 (2006), 539-571, math.QA/0501405.
  3. van de Bult F.J., Rains E.M., Stokman J.V., Properties of generalized univariate hypergeometric functions, Comm. Math. Phys. 275 (2007), 37-95, math.CA/0607250.
  4. Chalykh O.A., Macdonald polynomials and algebraic integrability, Adv. Math. 166 (2002), 193-259, math.QA/0212313.
  5. Chalykh O.A., Etingof P., Orthogonality relations and Cherednik identities for multivariable Baker-Akhiezer functions, arXiv:1111.0515.
  6. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
  7. Gupta D.P., Masson D.R., Contiguous relations, continued fractions and orthogonality, Trans. Amer. Math. Soc. 350 (1998), 769-808, math.CA/9511218.
  8. Haine L., Iliev P., Askey-Wilson type functions with bound states, Ramanujan J. 11 (2006), 285-329, math.QA/0203136.
  9. Ismail M.E.H., Rahman M., The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), 201-237.
  10. Koelink E., Stokman J.V., The Askey-Wilson function transform, Int. Math. Res. Not. (2001), no. 22, 1203-1227, math.CA/0004053.
  11. Koelink E., Stokman J.V., Fourier transforms on the quantum ${\rm SU}(1,1)$ group, Publ. Res. Inst. Math. Sci. 37 (2001), 621-715, math.QA/9911163.
  12. Koornwinder T., Comment on the paper "Macdonald polynomials and algebraic integrability" by O.A. Chalykh, available at
  13. Letzter G., Stokman J.V., Macdonald difference operators and Harish-Chandra series, Proc. Lond. Math. Soc. (3) 97 (2008), 60-96, math.QA/0701218.
  14. van Meer M., Bispectral quantum Knizhnik-Zamolodchikov equations for arbitrary root systems, Selecta Math. (N.S.) 17 (2011), 183-221, arXiv:0912.3784.
  15. van Meer M., Stokman J., Double affine Hecke algebras and bispectral quantum Knizhnik-Zamolodchikov equations, Int. Math. Res. Not. (2010), no. 6, 969-1040, arXiv:0812.1005.
  16. Noumi M., Stokman J.V., Askey-Wilson polynomials: an affine Hecke algebra approach, in Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004, 111-144, math.QA/0001033.
  17. Rahman M., The linearization of the product of continuous $q$-Jacobi polynomials, Canad. J. Math. 33 (1981), 961-987.
  18. Rahman M., Verma A., Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc. 335 (1993), 277-302.
  19. Ruijsenaars S.N.M., A generalized hypergeometric function satisfying four analytic difference equations of Askey-Wilson type, Comm. Math. Phys. 206 (1999), 639-690.
  20. Ruijsenaars S.N.M., Quadratic transformations for a function that generalizes ${}_2F_1$ and the Askey-Wilson polynomials, Ramanujan J. 13 (2007), 339-364.
  21. Sauloy J., Systèmes aux $q$-différences singuliers réguliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble) 50 (2000), 1021-1071.
  22. Singh V.N., The basic analogues of identities of the Cayley-Orr type, J. London Math. Soc. 34 (1959), 15-22.
  23. Slater L.J., A note on equivalent product theorems, Math. Gaz. 38 (1954), 127-128.
  24. Stokman J.V., An expansion formula for the Askey-Wilson function, J. Approx. Theory 114 (2002), 308-342, math.CA/0105093.
  25. Stokman J.V., The $c$-function expansion of a basic hypergeometric function associated to root systems, arXiv:1109.0613.
  26. Suslov S.K., Some orthogonal very-well-poised ${}_8\phi_7$-functions that generalize Askey-Wilson polynomials, Ramanujan J. 5 (2001), 183-218, math.CA/9707213.

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