# zbMATH — the first resource for mathematics

On a generalization of the Rogers generating function. (English) Zbl 1461.33007
Summary: We derive a generalization of the Rogers generating function for the continuous $$q$$-ultraspherical/Rogers polynomials whose coefficient is a $$_2\phi_1$$. From that expansion, we derive corresponding specialization and limit transition expansions for the continuous $$q$$-Hermite, continuous $$q$$-Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using a generalized expansion of the Rogers generating function in terms of the Askey-Wilson polynomials by Ismail & Simeonov whose coefficient is a $$_8\phi_7$$, we derive corresponding generalized expansions for the Wilson, continuous $$q$$-Jacobi, and Jacobi polynomials. By comparing the coefficients of the Askey-Wilson expansion to our continuous $$q$$-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric functions which relates an $$_8\phi_7$$ to a $$_2\phi_1$$. We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality.

##### MSC:
 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals
Full Text:
##### References:
 [1] Andrews, G. E.; Askey, R.; Roy, R., Special Functions, Encyclopedia Math. Appl., vol. 71, (1999), Cambridge University Press: Cambridge University Press Cambridge [2] Askey, R.; Ismail, M. E.H., A generalization of ultraspherical polynomials, (Studies in Pure Mathematics, (1983), Birkhäuser: Birkhäuser Basel), 55-78 · Zbl 0532.33006 [3] Brown, B. M.; Evans, W. D.; Ismail, M. E.H., The Askey-Wilson polynomials and q-Sturm-Liouville problems, Math. Proc. Cambridge Philos. Soc., 119, 1, 1-16, (1996) · Zbl 0860.33012 [4] Cohl, H. S., Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems, SIGMA Symmetry Integrability Geom. Methods Appl., 9, 042, 26, (2013) · Zbl 1270.35009 [5] Cohl, H. S., On a generalization of the generating function for Gegenbauer polynomials, Integral Transforms Spec. Funct., 24, 10, 807-816, (2013) · Zbl 1280.35003 [6] Cohl, H. S.; Dominici, D. E., Generalized Heine’s identity for complex Fourier series of binomials, Proc. Roy. Soc. Edinburgh Sect. A, 467, 333-345, (2011) · Zbl 1219.33008 [7] Cohl, H. S.; MacKenzie, C.; Volkmer, H., Generalizations of generating functions for hypergeometric orthogonal polynomials with definite integrals, J. Math. Anal. Appl., 407, 2, 211-225, (2013) · Zbl 1306.33016 [8] Durand, L.; Fishbane, P. M.; Simmons, L. M., Expansion formulas and addition theorems for Gegenbauer functions, J. Math. Phys., 17, 11, 1933-1948, (1976) · Zbl 0336.33004 [9] Fano, U.; Rau, A. R.P., Symmetries in Quantum Physics, (1996), Academic Press Inc.: Academic Press Inc. San Diego, CA [10] Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia Math. Appl., vol. 96, (2004), Cambridge University Press: Cambridge University Press Cambridge, With a foreword by Richard Askey · Zbl 1129.33005 [11] Gegenbauer, L., Über einige bestimmte Integrale, Sitz. Kaiserl. Akad. Wiss. Mat.-Natur. Cl., 70, 433-443, (1874) [12] Granovskiĭ, Ya. I.; Zhedanov, A. S., Spherical q-functions, J. Phys. A: Math. Gen., 26, 17, 4331-4338, (1993) · Zbl 0854.33013 [13] Groenevelt, W., The Wilson function transform, Int. Math. Res. Not., 2003, 52, 2779-2817, (2003) · Zbl 1079.33005 [14] Heine, E., Handbuch der Kugelfunctionen, Theorie und Anwendungen (vol. 1), (1878), Druck und Verlag von G. Reimer: Druck und Verlag von G. Reimer Berlin · JFM 10.0332.01 [15] Heine, E., Handbuch der Kugelfunctionen, Theorie und Anwendungen (vol. 2), (1881), Druck und Verlag von G. Reimer: Druck und Verlag von G. Reimer Berlin [16] Iorgov, N. Z.; Klimyk, A. U., The q-Laplace operator and q-harmonic polynomials on the quantum vector space, J. Math. Phys., 42, 3, 1326-1345, (2001) · Zbl 1023.33015 [17] Ismail, M. E.H., Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia Math. Appl., vol. 98, (2009), Cambridge University Press: Cambridge University Press Cambridge, With two chapters by Walter Van Assche. With a foreword by Richard A. Askey. Corrected reprint of the 2005 original · Zbl 1172.42008 [18] Ismail, M. E.H.; Mansour, Z. S.I., Functions of the second kind for classical polynomials, Adv. in Appl. Math., 54, 66-104, (2014) · Zbl 1285.33013 [19] Ismail, M. E.H.; Simeonov, P., Formulas and identities involving the Askey-Wilson operator, Adv. in Appl. Math., 76, 68-96, (2016) · Zbl 1336.33026 [20] Koekoek, R.; Lesky, P. A.; Swarttouw, R. F., Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer Monogr. Math., (2010), Springer-Verlag: Springer-Verlag Berlin, With a foreword by Tom H. Koornwinder · Zbl 1200.33012 [21] Koornwinder, T. H., Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators, SIGMA Symmetry Integrability Geom. Methods Appl., 11, (2015) · Zbl 1325.33003 [22] Koornwinder, T. H., Dual addition formula for continuous q-ultraspherical polynomials, (Proceedings of the 17th Annual Conference for the Society for Special Functions and Their Applications (SSFA), vol. 17, (2018)), 1-29 [23] Noumi, M.; Umeda, T.; Wakayama, M., Dual pairs, spherical harmonics and a Capelli identity in quantum group theory, Compos. Math., 104, 3, 227-277, (1996) · Zbl 0930.17012 [24] Release 1.0.21 of 2018-12-15 [25] Rahman, M., An integral representation of a $${}_{10}\varphi_9$$ and continuous bi-orthogonal $${}_{10}\varphi_9$$ rational functions, Canad. J. Math., 38, 3, 605-618, (1986) · Zbl 0599.33015 [26] Rahman, M.; Verma, A., Quadratic transformation formulas for basic hypergeometric series, Trans. Amer. Math. Soc., 335, 1, 277-302, (1993) · Zbl 0767.33011 [27] Rains, E. M.; Warnaar, S. O., Bounded Littlewood Identities, Mem. Amer. Math. Soc., (2018), in press [28] Rogers, L. J., On the expansion of some infinite products, Proc. Lond. Math. Soc., 24, 337-352, (1893) · JFM 25.0432.01 [29] Szegő, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, (1959), American Mathematical Society: American Mathematical Society Providence, R.I. · JFM 61.0386.03 [30] Wen, Z. Y.; Avery, J., Some properties of hyperspherical harmonics, J. Math. Phys., 26, 3, 396-403, (1985) · Zbl 0575.33006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.