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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
Published electronically: October 14, 2011
MathSciNet review: 2888193
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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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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

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

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

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