Two classes of special functions using Fourier transforms of generalized ultraspherical and generalized Hermite polynomials

Authors:
Mohammad Masjed-Jamei and Wolfram Koepf

Journal:
Proc. Amer. Math. Soc. **140** (2012), 2053-2063

MSC (2010):
Primary 33C45, 42A38

DOI:
https://doi.org/10.1090/S0002-9939-2011-11063-3

Published electronically:
October 14, 2011

MathSciNet review:
2888193

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Abstract | References | Similar Articles | Additional Information

Abstract: Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. Motivated by the paper [H. T. Koelink, On Jacobi and continuous Hahn polynomials, *Proc. Amer. Math. Soc.*, **124** (1996) 887-898], in this paper we introduce two new classes of orthogonal functions, which are respectively Fourier transforms of the generalized ultraspherical polynomials and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval's identity.

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Additional Information

**Mohammad Masjed-Jamei**

Affiliation:
Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

Email:
mmjamei@kntu.ac.ir, mmjamei@yahoo.com

**Wolfram Koepf**

Affiliation:
Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany

Email:
koepf@mathematik.uni-kassel.de

DOI:
https://doi.org/10.1090/S0002-9939-2011-11063-3

Received by editor(s):
August 17, 2010

Received by editor(s) in revised form:
January 10, 2011, and February 10, 2011

Published electronically:
October 14, 2011

Additional Notes:
This research was supported in part by a grant from IPM, No. 89330021, and by a grant from “Bonyade Mellie Nokhbegan”, No. PM/1184

Communicated by:
Walter Van Assche

Article copyright:
© Copyright 2011
By the authors