Hostname: page-component-5cf477f64f-54txb Total loading time: 0 Render date: 2025-04-05T05:25:42.964Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalization of Gasper's Kernel for Hahn Polynomials: Application to Pollaczek Polynomials

Published online by Cambridge University Press:  20 November 2018

Mizan Rahman
Mizan Rahman*
Affiliation:
Carleton University, Ottawa, Ontario
Rights & Permissions [Opens in a new window]

Extract

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.

In this paper we consider a generalization of the discrete Poisson kernel for the Hahn polynomials obtained recently by Gasper [6]. The Hahn polynomials of degree n are defined by

(1.1)

and are known to be orthogonal on the set of non-negative integers x = 0, 1, . . . , N provided Re α, β > - l or Re α, β < -N [7; 8].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 1 , 01 February 1978 , pp. 133 - 146
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Askey, R., Orthogonal polynomials and special functions, Vol. 21, SIAM series of Regional Conference Lectures (1974).Google Scholar
2. Bailey, W. N., Generalized hyper geometric series (Stechert-Hafner Serivce Agency, Inc., New York, 1964).Google Scholar
3. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. H. eds., Higher transcendental junctions, Bateman Manuscript Project, Vol. I (McGraw-Hill, 1953).Google Scholar
4. A., Erdélyi, Magnus, W., Higher transcendental junctions, Bateman Manuscript Project, Vol. II (McGraw- Hill, 1953).Google Scholar
5. Fields, J. L. and Ismail, M. E. H., Polynomial expansions, Mathematics of Computation 20 (1975), 894–902.Google Scholar
6. Gasper, G., Nonnegativity of a discrete Poisson kernel for the Hahn polynomials, J. Math. Anal. Appl. 42 (1973), 438–451.Google Scholar
7. Gasper, G., Positivity and special functions, in Theory and application of special functions, ed. Askey, Richard A. (Academic Press, 1975).Google Scholar
8. Karlin, S. and McGregor, J., The Hahn polynomials, formulas and an application, Scripta Math. 26 (1961), 33–46.Google Scholar
9. Mathews, J. and Walker, R. L., Mathematical methods of physics, 2nd edn. (W. A. Benjamin, Inc., New York, 1970).Google Scholar
10. Mizan, Rahman, On a generalization of the Poisson kernel for Jacobi polynomials, SIAM J. Math. Anal., to appear.Google Scholar
11. Slater, L. J., Generalized hyper geometric functions (Cambridge University Press, 1966).Google Scholar
12. Srivastava, H. M., Infinite series of certain products involving AppelVs double hyper geometric functions, Glasnik Mat. 4 (24) (1969), 67–73.Google Scholar
13. Szego, G., Orthogonal polynomials, American Mathematical Society Colloquium publications, Vol. XXIII, 4th edn. (1975).Google Scholar
14. Verma, A., Some transformations of series with arbitrary terms, 1st Lombardo Accad. Sci. Lett. Rend. A 106 (1972), 342–353.Google Scholar
15. Zemanian, A. H., Distribution theory and transform analysis (McGraw-Hill Book Company, 1965).Google Scholar