A Second Addition Formula for Continuous q-Ultraspherical Polynomials

  • Tom H. Koornwinder
Part of the Developments in Mathematics book series (DEVM, volume 13)


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.


Quantum Group Addition Formula Connection Formula Ultraspherical Polynomial Wilson Polynomial 
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  1. Askey, R. (1975). Orthogonal polynomials and special functions, volume 21 of Regional Conference Series in Applied Math. SIAM, Philadelphia, PA.Google Scholar
  2. 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.Google Scholar
  3. Askey, R. and Roy, R. (1986). More q-beta integrals. Rocky Mountain J. Math., 16:365–372.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Askey, R. and Wilson, J. (1985). Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319):iv+55.MathSciNetGoogle Scholar
  5. Berndt, B. C., editor (1991). Ramanujan's notebooks, Part III. Springer-Verlag, New York.zbMATHGoogle Scholar
  6. Bustoz, J. and Suslov, S. K. (1998). Basic analog of Fourier series on a q-quadratic grid. Methods Appl. Anal., 5:1–38.MathSciNetzbMATHGoogle Scholar
  7. De Sole, A. and Kac, V. (2003). On integral representations of q-gamma and q-beta functions. Preprint: arXiv:math.QA/0302032.Google Scholar
  8. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). Higher transcendental functions, Vol. II. McGraw-Hill.Google Scholar
  9. Gasper, G. (1984). Letter to R. Askey dated July 23, 1984. Personal communication.Google Scholar
  10. Gasper, G. (1987). Solution to problem #6497 (q-analogues of a gamma function identity, by R. Askey). Amer. Math. Monthly, 94:199–201.CrossRefMathSciNetGoogle Scholar
  11. Gasper, G. and Rahman, M. (1990). Basic hypergeometric series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  12. 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.Google Scholar
  13. Ismail, M. E. H. and Masson, D. R. (1995). Generalized orthogonality and continued fractions. J. Approx. Theory, 83:1–40.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 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.Google Scholar
  15. Ismail, M. E. H. and Zhang, R. (1994). Diagonalization of certain integral operators. Adv. Math., 109:1–33.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 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: Scholar
  17. 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.Google Scholar
  18. 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.Google Scholar
  19. Koelink, H. T. (1991). On quantum groups and q-special functions. PhD thesis, University of Leiden.Google Scholar
  20. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  21. Koornwinder, T. H. (1993). Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal., 24:795–813.CrossRefzbMATHMathSciNetGoogle Scholar
  22. Pastro, P. I. (1985). Orthogonal polynomials and some q-beta integrals of Ramanujan. J. Math. Anal. Appl., 112:517–540.CrossRefzbMATHMathSciNetGoogle Scholar
  23. Rahman, M. (1986). An integral representation of a 10ϕ9 and continuous bi-orthogonal 10ϕ9 rational functions. Canad. J. Math., 38:605–618.zbMATHMathSciNetGoogle Scholar
  24. Rahman, M. (1988). An addition theorem and some product formulas for q-Bessel functions. Canad. J. Math., 45:1203–1221.Google Scholar
  25. Rahman, M. and Verma, A. (1986). Product and addition formula for the continuous q-ultraspherical polynomials. SIAM J. Math. Anal., 17:1461–1474.CrossRefMathSciNetzbMATHGoogle Scholar
  26. 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.Google Scholar
  27. Spiridonov, V. and Zhedanov, A. (2000). Spectral transformation chains and some new biorthogonal rational functions. Comm. Math. Phys., 210:49–83.CrossRefMathSciNetzbMATHGoogle Scholar
  28. Wilson, J. A. (1991). Orthogonal functions from Gram determinants. SIAM J. Math. Anal., 22:1147–1155.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Tom H. Koornwinder
    • 1
  1. 1.Korteweg-de Vries InstituteUniversity of AmsterdamAmsterdamThe Netherlands

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