Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 
 

 

Two classes of special functions using Fourier transforms of generalized ultraspherical and generalized Hermite polynomials


Authors: Mohammad Masjed-Jamei and Wolfram Koepf
Journal: Proc. Amer. Math. Soc. 140 (2012), 2053-2063
MSC (2010): Primary 33C45, 42A38
DOI: https://doi.org/10.1090/S0002-9939-2011-11063-3
Published electronically: October 14, 2011
MathSciNet review: 2888193
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. Motivated by the paper [H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996) 887-898], in this paper we introduce two new classes of orthogonal functions, which are respectively Fourier transforms of the generalized ultraspherical polynomials and generalized Hermite polynomials, and then obtain their orthogonality relations using Parseval's identity.


References [Enhancements On Off] (What's this?)

  • 1. W. Al-Salam, W. R. Allaway and R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc., 284 (1984) 39-55. MR 742411 (85j:33005)
  • 2. R. Askey, An integral of Ramanujan and orthogonal polynomials, J. Indian Math. Soc. 51 (1987) 27-36. MR 988306 (90d:33004)
  • 3. R. Askey, Orthogonal polynomials old and new, and some combinatorial connections, Enumeration and Design, (D.M. Jacson and S.A. Vanstone, eds.), Academic Press, New York, 1984, 67-84. MR 782309 (87a:05022)
  • 4. W. N. Bailey, Generalized Hypergeometric Series. Cambridge Tracts 32, Cambridge University PFTV, 1935. Reprinted by Hafner Publishing Company, 1972. MR 0185155 (32:2625)
  • 5. T. S. Chihara, Introduction to Orthogonal Polynomials, Gordon & Breach, New York, 1978. MR 0481884 (58:1979)
  • 6. Digital Library of Mathematical Functions, DLMF available at http://dlmf.nist.gov/
  • 7. A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. Vol. 2, McGraw-Hill, 1954. MR 0065685 (16:468c)
  • 8. M. E. H. Ismail, On sieved orthogonal polynomials III: Orthogonality on several intervals, Trans. Amer. Math. Soc., 294 (1986) 89-111. MR 819937 (87j:33014b)
  • 9. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encycl. of Math. 98, Cambridge Univ. Press, Cambridge, 2005. MR 2191786 (2007f:33001)
  • 10. H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc., 124 (1996) 887-898. MR 1307541 (96f:33018)
  • 11. W. Koepf, Hypergeometric Summation. Braunschweig/Wiesbaden, Vieweg, 1988. MR 1644447 (2000c:33002)
  • 12. W. Koepf and M. Masjed-Jamei, Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials, Proc. Amer. Math. Soc., 135 (2007) 3599-3606. MR 2336575 (2008j:33004)
  • 13. 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 in Math. 1171, Springer, 1985, 174-183. MR 838982 (87g:33007)
  • 14. M. Masjed-Jamei, A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions, J. Math. Anal. Appl., 325 (2007) 753-775. MR 2270049 (2008c:33008)
  • 15. M. Masjed-Jamei, Biorthogonal exponential sequences with weight function on the real line and an orthogonal sequence of trigonometric functions, Proc. Amer. Math. Soc., 136 (2008) 409-417. MR 2358478 (2009a:42040)
  • 16. L. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895) 15-32.
  • 17. G. Szegő, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Providence, RI, 1975. MR 0372517 (51:8724)
  • 18. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1927. MR 1424469 (97k:01072)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 33C45, 42A38

Retrieve articles in all journals with MSC (2010): 33C45, 42A38


Additional Information

Mohammad Masjed-Jamei
Affiliation: Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran – and – School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran
Email: mmjamei@kntu.ac.ir, mmjamei@yahoo.com

Wolfram Koepf
Affiliation: Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany
Email: koepf@mathematik.uni-kassel.de

DOI: https://doi.org/10.1090/S0002-9939-2011-11063-3
Received by editor(s): August 17, 2010
Received by editor(s) in revised form: January 10, 2011, and February 10, 2011
Published electronically: October 14, 2011
Additional Notes: This research was supported in part by a grant from IPM, No. 89330021, and by a grant from “Bonyade Mellie Nokhbegan”, No. PM/1184
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 By the authors

American Mathematical Society