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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[AAR]**George E. Andrews, Richard Askey, and Ranjan Roy,*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958****[Askey1]**Richard Askey,*Continuous Hahn polynomials*, J. Phys. A**18**(1985), no. 16, L1017–L1019. MR**812420****[Askey2]**Richard Askey,*An integral of Ramanujan and orthogonal polynomials*, J. Indian Math. Soc. (N.S.)**51**(1987), 27–36 (1988). MR**988306****[AS]**N. M. Atakishiyev and S. K. Suslov,*The Hahn and Meixner polynomials of an imaginary argument and some of their applications*, J. Phys. A**18**(1985), no. 10, 1583–1596. MR**796065****[Bail]**W. N. Bailey,*Generalized hypergeometric series*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR**0185155****[EMOT]**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,*Tables of integral transforms. Vol. II*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR**0065685****[Koel]**H. T. Koelink,*On Jacobi and continuous Hahn polynomials*, Proc. Amer. Math. Soc.**124**(1996), no. 3, 887–898. MR**1307541**, https://doi.org/10.1090/S0002-9939-96-03190-5**[Koep]**Wolfram Koepf,*Hypergeometric summation*, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1998. An algorithmic approach to summation and special function identities. MR**1644447****[Koor1]**T. H. Koornwinder,*Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform*, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 174–183. MR**838982**, https://doi.org/10.1007/BFb0076542**[Koor2]**Tom H. Koornwinder,*Group theoretic interpretations of Askey’s scheme of hypergeometric orthogonal polynomials*, Orthogonal polynomials and their applications (Segovia, 1986) Lecture Notes in Math., vol. 1329, Springer, Berlin, 1988, pp. 46–72. MR**973421**, https://doi.org/10.1007/BFb0083353**[Les]**P. Lesky,*Eine Charakterisierung der klassischen kontinuierlichen, diskreten und -Orthogonalpolynome*, Shaker, Aachen, 2005.**[Mas1]**Mohammad Masjed Jamei,*Classical orthogonal polynomials with weight function ((𝑎𝑥+𝑏)²+(𝑐𝑥+𝑑)²)^{-𝑝}exp(𝑞𝐴𝑟𝑐𝑡𝑔((𝑎𝑥+𝑏)/(𝑐𝑥+𝑑))), 𝑥∈(-∞,∞) and a generalization of 𝑇 and 𝐹 distributions*, Integral Transforms Spec. Funct.**15**(2004), no. 2, 137–153. MR**2053407**, https://doi.org/10.1080/10652460410001663456**[Mas2]**Mohammad Masjedjamei,*Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation*, Integral Transforms Spec. Funct.**13**(2002), no. 2, 169–191. MR**1915513**, https://doi.org/10.1080/10652460212898**[PFTV]**William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery,*Numerical recipes in FORTRAN*, 2nd ed., Cambridge University Press, Cambridge, 1992. The art of scientific computing; With a separately available computer disk. MR**1196230**

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery,*Numerical recipes in C*, 2nd ed., Cambridge University Press, Cambridge, 1992. The art of scientific computing. MR**1201159****[Rama]**S. Ramanujan,*A class of definite integrals*, Quarterly J. Math.**48**(1920), 294-310.**[WF]**John E. Freund and Ronald E. Walpole,*Mathematical statistics*, 3rd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR**591029****[WW]**E. T. Whittaker and G. N. Watson,*A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions*, Fourth edition. Reprinted, Cambridge University Press, New York, 1962. MR**0178117**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
33C45

Retrieve articles in all journals with MSC (2000): 33C45

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