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Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials

Authors: Wolfram Koepf and Mohammad Masjed-Jamei
Journal: Proc. Amer. Math. Soc. 135 (2007), 3599-3606
MSC (2000): Primary 33C45
Published electronically: June 29, 2007
MathSciNet review: 2336575
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Abstract: Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by $ \exp(-x^2/2)$, which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

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Additional Information

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

Mohammad Masjed-Jamei
Affiliation: Department of Mathematics, K. N. Toosi University of Technology, Sayed Khandan, Jolfa Av., Tehran, Iran

Keywords: Classical orthogonal polynomials, Fourier transform, hypergeometric functions, Gosper identity, Ramanujan integral
Received by editor(s): January 1, 2006
Received by editor(s) in revised form: August 16, 2006
Published electronically: June 29, 2007
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society