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

DOI:
https://doi.org/10.1090/S0002-9939-07-08889-2

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 , 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

Email:
koepf@mathematik.uni-kassel.de

**Mohammad Masjed-Jamei**

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

Email:
mmjamei@yahoo.com

DOI:
https://doi.org/10.1090/S0002-9939-07-08889-2

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