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]**G. E. Andrews, R. Askey and R. Roy,*Special Functions*, Encyclopedia of Mathematics and its Applications**71**, Cambridge University Press, Cambridge, 1999. MR**1688958 (2000g:33001)****[Askey1]**R. Askey,*Continuous Hahn polynomials*, J. Physics A**18**, 1985, L1017-L1019. MR**0812420 (87d:33021)****[Askey2]**R. Askey,*An integral of Ramanujan and orthogonal polynomials*, J. Indian Math. Soc.**51**, 1987, 27-36. MR**0988306 (90d:33004)****[AS]**N. M. Atakishiyev and S. K. Suslov,*The Hahn and Meixner polynomials of an imaginary argument and some of their applications*, J. Physics A**18**, 1985, 1583-1596. MR**0796065 (87i:33021)****[Bail]**W. N. Bailey,*Generalized Hypergeometric Series*, Cambridge Tracts 32, Cambridge University PFTV, 1935. Reprinted by Hafner Publishing Company, 1972. MR**0185155 (32:2625)****[EMOT]**A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,*Tables of Integral Transforms*, Vol. 2, McGraw-Hill, 1954. MR**0065685 (16:468c)****[Koel]**H. T. Koelink,*On Jacobi and continuous Hahn polynomials*, Proc. Amer. Math. Soc.**124**, 1996, 887-898. MR**1307541 (96f:33018)****[Koep]**W. Koepf,*Hypergeometric Summation*, Braunschweig/Wiesbaden, Vieweg, 1998. MR**1644447 (2000c:33002)****[Koor1]**T. H. Koornwinder,*Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transform*, Polynômes Orthogonaux et Applications (C. Brezinski, A. Draux, A. P. Magnus, P. Maroni and A. Ronveaux, Eds.), Lecture Notes Math. 1171, Springer, 1985, 174-183. MR**0838982 (87g:33007)****[Koor2]**T. H. Koornwinder,*Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials*, Orthogonal Polynomials and their Applications (M. Alfaro, J. S. Dehesa, F. J. Marcellan, J. L. Rubio de Francia and J. Vinuesa, Eds.), Lecture Notes Math. 1329, Springer, 1988, 46-72. MR**0973421 (90b:33024)****[Les]**P. Lesky,*Eine Charakterisierung der klassischen kontinuierlichen, diskreten und -Orthogonalpolynome*, Shaker, Aachen, 2005.**[Mas1]**M. Masjed-Jamei,*Classical orthogonal polynomials with weight function ; and a generalization of and distributions*, J. Integral Transforms and Special Functions**15**(2), 2004, 137-153. MR**2053407 (2005b:33011)****[Mas2]**M. Masjed-Jamei,*Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation*, J. Integral Transforms and Special Functions**13**(2), 2002, 169-190. MR**1915513 (2003i:33011)****[PFTV]**W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling,*Beta Function, Student Distribution and -Distribution*, Section 6.2 in*Numerical Recipes in Fortran: The Art of Scientific Computing*, second edition, Cambridge University Press, Cambridge, 1992, 219-223. MR**1196230 (93i:65001a)****[Rama]**S. Ramanujan,*A class of definite integrals*, Quarterly J. Math.**48**(1920), 294-310.**[WF]**R. E. Walpole and J. E. Freund,*Mathematical Statistics*, Prentice-Hall, 1980. MR**0591029 (81k:62002)****[WW]**E. T. Whittaker and G. N. Watson,*A Course of Modern Analysis*, 4th ed., Cambridge University Press, Cambridge, 1962. MR**0178117 (31:2375)**

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