Some remarks on very-well-poised ${}_8\phi_7$ series

Jasper V. Stokman

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

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.

Détail

Keywords: very-well-poised basic hypergeometric series, Askey–Wilson functions, quadratic transformation formulas, theta functions.

Bibliographie

[1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54, no. 319, 1985 | MR

[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 ; arXiv: math.QA/0501405 | DOI | MR | Zbl

[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 ; arXiv: math.CA/0607250 | DOI | MR | Zbl

[4] Chalykh O.A., “Macdonald polynomials and algebraic integrability”, Adv. Math., 166 (2002), 193–259 ; arXiv: math.QA/0212313 | DOI | MR | Zbl

[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, 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI | MR | Zbl

[7] Gupta D.P., Masson D.R., “Contiguous relations, continued fractions and orthogonality”, Trans. Amer. Math. Soc., 350 (1998), 769–808 ; arXiv: math.CA/9511218 | DOI | MR | Zbl

[8] Haine L., Iliev P., “Askey–Wilson type functions with bound states”, Ramanujan J., 11 (2006), 285–329 ; arXiv: math.QA/0203136 | DOI | MR | Zbl

[9] Ismail M.E.H., Rahman M., “The associated Askey–Wilson polynomials”, Trans. Amer. Math. Soc., 328 (1991), 201–237 | DOI | MR | Zbl

[10] Koelink E., Stokman J.V., “The Askey–Wilson function transform”, Int. Math. Res. Not., 2001, no. 22, 1203–1227 ; arXiv: math.CA/0004053 | DOI | MR | Zbl

[11] Koelink E., Stokman J.V., “Fourier transforms on the quantum $\mathrm{SU}(1,1)$ group”, Publ. Res. Inst. Math. Sci., 37 (2001), 621–715 ; arXiv: math.QA/9911163 | DOI | MR | Zbl

[12] Koornwinder T., Comment on the paper “Macdonald polynomials and algebraic integrability” by O.A. Chalykh available at http://staff.science.uva.nl/~thk/art/comment/ChalykhComment.pdf

[13] Letzter G., Stokman J.V., “Macdonald difference operators and Harish-Chandra series”, Proc. Lond. Math. Soc. (3), 97 (2008), 60–96 ; arXiv: math.QA/0701218 | DOI | MR | Zbl

[14] van Meer M., “Bispectral quantum Knizhnik–Zamolodchikov equations for arbitrary root systems”, Selecta Math. (N.S.), 17 (2011), 183–221 ; arXiv: 0912.3784 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[16] Noumi M., Stokman J.V., “Askey–Wilson polynomials: an affine Hecke algebra approach”, Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY, 2004, 111–144 ; arXiv: math.QA/0001033 | MR | Zbl

[17] Rahman M., “The linearization of the product of continuous $q$-Jacobi polynomials”, Canad. J. Math., 33 (1981), 961–987 | DOI | MR | Zbl

[18] Rahman M., Verma A., “Quadratic transformation formulas for basic hypergeometric series”, Trans. Amer. Math. Soc., 335 (1993), 277–302 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[21] Sauloy J., “Systèmes aux $q$-différences singuliers réguliers: classification, matrice de connexion et monodromie”, Ann. Inst. Fourier (Grenoble), 50 (2000), 1021–1071 | DOI | MR | Zbl

[22] Singh V.N., “The basic analogues of identities of the Cayley–Orr type”, J. London Math. Soc., 34 (1959), 15–22 | DOI | MR | Zbl

[23] Slater L.J., “A note on equivalent product theorems”, Math. Gaz., 38 (1954), 127–128 | DOI

[24] Stokman J.V., “An expansion formula for the Askey–Wilson function”, J. Approx. Theory, 114 (2002), 308–342 ; arXiv: math.CA/0105093 | DOI | MR | Zbl

[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 ; arXiv: math.CA/9707213 | DOI | MR | Zbl