Abstract
This paper provides the details of Remark 5.4 in the author's paper “Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group,” SIAM J. Math. Anal. 24 (1993), 795–813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finite Fourier series with a product of two 3ϕ2's as Fourier coefficients. The proof given there used the quantum group interpretation. Here this identity will be generalized to a 3-parameter class of Askey-Wilson polynomials being expanded in terms of continuous q-ultraspherical polynomials with a product of two 2ϕ2's as coefficients, and an analytic proof will be given for it. Then Gegenbauer's addition formula for ultraspherical polynomials and Rahman's addition formula for q-Bessel functions will be obtained as limit cases. This q-analogue of Gegenbauer's addition formula is quite different from the addition formula for continuous q-ultraspherical polynomials obtained by Rahman and Verma in 1986. Furthermore, the functions occurring as factors in the expansion coefficients will be interpreted as a special case of a system of biorthogonal rational functions with respect to the Askey-Roy q-beta measure. A degenerate case of this biorthogonality are Pastro's biorthogonal polynomials associated with the Stieltjes-Wigert polynomials.
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References
Askey, R. (1975). Orthogonal polynomials and special functions, volume 21 of Regional Conference Series in Applied Math. SIAM, Philadelphia, PA.
Askey, R. (1988). Beta integrals and q-extensions. In Balakrishnan, R., Padmanabhan, K. S., and Thangaraj, V., editors, Proceedings of the Ramanujan Centennial International Conference (Annamalainagar, 1987), volume 1 of RMS Publ., pages 85–102, Annamalainagar. Ramanujan Mathematical Society.
Askey, R. and Roy, R. (1986). More q-beta integrals. Rocky Mountain J. Math., 16:365–372.
Askey, R. and Wilson, J. (1985). Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319):iv+55.
Berndt, B. C., editor (1991). Ramanujan's notebooks, Part III. Springer-Verlag, New York.
Bustoz, J. and Suslov, S. K. (1998). Basic analog of Fourier series on a q-quadratic grid. Methods Appl. Anal., 5:1–38.
De Sole, A. and Kac, V. (2003). On integral representations of q-gamma and q-beta functions. Preprint: arXiv:math.QA/0302032.
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). Higher transcendental functions, Vol. II. McGraw-Hill.
Gasper, G. (1984). Letter to R. Askey dated July 23, 1984. Personal communication.
Gasper, G. (1987). Solution to problem #6497 (q-analogues of a gamma function identity, by R. Askey). Amer. Math. Monthly, 94:199–201.
Gasper, G. and Rahman, M. (1990). Basic hypergeometric series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge.
Hardy, G. H. (1915). Proof of a formula of Mr. Ramanujan. Messenger Math., 44:18–21. Reprinted in Collected Papers of G. H. Hardy, Vol. 5, Oxford, 1972, pp. 594–597.
Ismail, M. E. H. and Masson, D. R. (1995). Generalized orthogonality and continued fractions. J. Approx. Theory, 83:1–40.
Ismail, M. E. H., Masson, D. R., and Suslov, S. K. (1999). The q-Bessel function on a q-quadratic grid. In van Diejen, J. F. and Vinet, L., editors, Algebraic methods and q-special functions (Montréal, QC, 1996), volume 22 of CRM Proc. Lecture Notes, pages 183–200. Amer. Math. Soc., Providence, RI.
Ismail, M. E. H. and Zhang, R. (1994). Diagonalization of certain integral operators. Adv. Math., 109:1–33.
Koekoek, R. and Swarttouw, R. F. (1998). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17, Faculty of Technical Mathematics and Informatics, Delft University of Technology. Electronic version: http://aw.twi.tudelft.nl/~koekoek/askey/.
Koelink, E. (1997). Addition formulas for q-special functions. In Ismail, M. E. H., Masson, D. R., and Rahman, M., editors, Special functions, q-series and related topics (Toronto, ON, 1995), volume 14 of Fields Institute Communications, pages 109–129. Amer. Math. Soc., Providence, RI. Electronic version: arXiv:math.QA/9506216.
Koelink, E. and Stokman, J. V. (2001). The Askey-Wilson function transform scheme. In Bustoz, J., Ismail, M. E. H., and Suslov, S. K., editors, Special Functions 2000: Current perspective and future directions (Tempe, AZ, 2000), volume 30 of NATO Science Series II, pages 221–241. Kluwer Academic Publishers, Dordrecht. Electronic version: arXiv:math.CA/9912140.
Koelink, H. T. (1991). On quantum groups and q-special functions. PhD thesis, University of Leiden.
Koelink, H. T. (1994). The addition formula for continuous q-Legendre polynomials and associated spherical elements on the SU(2) quantum group related to Askey-Wilson polynomials. SIAM J. Math. Anal., 25:197–217.
Koornwinder, T. H. (1993). Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal., 24:795–813.
Pastro, P. I. (1985). Orthogonal polynomials and some q-beta integrals of Ramanujan. J. Math. Anal. Appl., 112:517–540.
Rahman, M. (1986). An integral representation of a 10ϕ9 and continuous bi-orthogonal 10ϕ9 rational functions. Canad. J. Math., 38:605–618.
Rahman, M. (1988). An addition theorem and some product formulas for q-Bessel functions. Canad. J. Math., 45:1203–1221.
Rahman, M. and Verma, A. (1986). Product and addition formula for the continuous q-ultraspherical polynomials. SIAM J. Math. Anal., 17:1461–1474.
Ramanujan, S. (1915). Some definite integrals. Messenger Math., 44:10–18. Reprinted in Collected papers of Srinivasa Ramanujan, Cambridge University Press, 1927; Chelsea, New York, 1962.
Spiridonov, V. and Zhedanov, A. (2000). Spectral transformation chains and some new biorthogonal rational functions. Comm. Math. Phys., 210:49–83.
Wilson, J. A. (1991). Orthogonal functions from Gram determinants. SIAM J. Math. Anal., 22:1147–1155.
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Koornwinder, T.H. (2005). A Second Addition Formula for Continuous q-Ultraspherical Polynomials. In: Ismail, M.E., Koelink, E. (eds) Theory and Applications of Special Functions. Developments in Mathematics, vol 13. Springer, Boston, MA. https://doi.org/10.1007/0-387-24233-3_14
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