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

Koepf, Wolfram; Masjed-Jamei, Mohammad

doi:10.1090/S0002-9939-07-08889-2

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials
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by Wolfram Koepf and Mohammad Masjed-Jamei PDF

Proc. Amer. Math. Soc. 135 (2007), 3599-3606 Request permission

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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