Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T13:08:55.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonnegative Linearization and Quadratic Transformation of Askey-Wilson Polynomials

Published online by Cambridge University Press:  20 November 2018

Ryszard Szwarc
Ryszard Szwarc*
Affiliation:
Institute of Mathematics, Wroclaw University, pi. Grunwaldzki 2/4, 50-384 Wroclaw, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Nonnegative product linearization of the Askey-Wilson polynomials is shown for a wide range of parameters. As a corollary we obtain Rahman's result on the continuous q-Jacobi polynomials with α ≥ β > — 1 and α + β + 1 ≥ 0.

Keywords

33D45orthogonal polynomialsAskey-Wilson polynomialsnonnegative linearization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 2 , 01 June 1996 , pp. 241 - 249
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Askey, R., Linearization of the product of orthogonal polynomials, in Problems in Analysis, R. Gunning, éd., Princeton University Press, Princeton, New Jersey, 1970, 223–228.Google Scholar
2. Askey, R., Orthogonal polynomials and special functions, Regional Conference Series in Applied Mathematics 21, Society for Industrial and Applied Mathematics, Philadelphia, Pensylwania, 1975.Google Scholar
3. Askey, R. and Wilson, J. A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54( 1985).Google Scholar
4. Dougall, J., A theorem ofSonine in Bessel functions, with two extensions to spherical harmonics, Proc. Edinburgh Math. Soc. 37(1919), 33–47.Google Scholar
5. Gasper, G., Linearization of the product of Jacobi polynomials, I, Canad. J. Math. 22(1970), 171—175.Google Scholar
6. Gasper, G., Linearization of the product of Jacobi polynomials. II, ibid., 22(1970), 582—593.Google Scholar
7. Gasper, G., A convolution structure and positivity of a generalized translation operator for the continuous q-Jacobi polynomials, in Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth International Group, Belmont, Calif. (1983), 44-59.Google Scholar
8. Gasper, G. and Rahman, M., Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge 35(1990).Google Scholar
9. Koornwinder, T., Discrete hypergroups associated with compact quantum Gelfand pairs, in Applications of Hypergroups and Related Measures Algebras, (eds. W. Connert, M. O. Gebuhrer and A. L. Schwartz), Contemporary Mathematics.Google Scholar
10. Rahman, M., The linearization of the product of continuous q-Jacobi polynomials, Can. J. Math. 33(1981), 961–987.Google Scholar
11. Szwarc, R., Orthogonal polynomials and a discrete boundary value problem, I, II, SIAM J. Math. Anal. 23(1992), 959–964, 965-969.Google Scholar
12. Szwarc, R., Nonnegative linearization and q-ultraspherical polynomials, Methods Appl. Anal., to appear.Google Scholar