Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-01T03:14:32.895Z Has data issue: false hasContentIssue false
  • English
  • Français

An Addition Theorem and Some Product Formulas for the Hahn-Exton q-Bessel Functions

Published online by Cambridge University Press:  20 November 2018

René F. Swarttouw
René F. Swarttouw*
Affiliation:
Delft University of Technology, Faculty of Technical Mathematicsand Informatics, Mekelweg 4, 2628 CD Delft, The Netherlands
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.

In this paper a q-analogue of Gegenbauer's addition formula for Bessel functions is obtained by using the orthogonality relation for the q-Ultraspherical polynomials of Rogers'. Also some product formulas and an integral representation for the Hahn-Exton q-Bessel functions are obtained.

Keywords

33D1533D45Hahn-Exton q-Bessel functionGegenbauer's addition formulaq-Ultraspherical polynomialsproduct formulasintegral representation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 867 - 879
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Askey, R. and H, M.E.. Ismail, A generalization of Ultrasphericalpolynomials. In: Studies in pure mathematics, Birkhäuser-Verlag, Basel, (1983), 55–78.Google Scholar
2. Exton, H., q-Hyper geometric Functions and Applications, Ellis Horwood, 1983.Google Scholar
3. Gasper, G. and Rahman, M., Basic Hyper geometric Series, Cambridge University Press, 1990.Google Scholar
4. Hahn, W., Beiträge zur Théorie der Heineschen Reihen, Math. Nachr., 2 (1949), 340–379.Google Scholar
5. Hahn, W., Die mechanische Deutung einer geometrischen Dijferenzengleichung, Z. Angew. Math. Mech., 33 (1953), 270–272.Google Scholar
6. H, M.E.. Ismail, The zeros of basic Bessel functions, the functions Jv+ax(x) and associated orthogonal polynomials, J. Math. Anal. Appl., 86 (1982), 1–19.Google Scholar
7. Koornwinder, T.H. and F, R. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. (to appear).Google Scholar
8. Rahman, M., An addition theorem and some product formulas for q-Bessel functions, Canad. J. Math., 40 (1988), 1203–1221.Google Scholar
9. Watson, G.N., A treatise on the theory of Bessel functions, Cambridge University Press, 1966.Google Scholar
10. Zygmund, A., Trigonometric Series, Cambridge University Press, 1968.Google Scholar