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q-Hermite Polynomials and Classical Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

Christian Berg  and
Mourad E. H. Ismail
Christian Berg
Affiliation:
Mathematics Institute Copenhagen University Universitetsparken5 DK-2100 Copenhagen Ø Denmark, e-mail: berg@math.ku.dk
Mourad E. H. Ismail
Affiliation:
Department of Mathematics University of South Florida Tampa, Florida 33620-5700 USA., e-mail: ismail@math.usf.edu
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Abstract

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We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

Keywords

33D4533A6544A60Askey-Wilson polynomialsq-orthogonal polynomialsorthogonality relationsq-beta integrals, q-Hermite polynomialsbiorthogonal rational functions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 1 , 01 February 1996 , pp. 43 - 63
Copyright
Copyright © Canadian Mathematical Society 1997

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