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
- [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
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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