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)



On Jacobi and continuous Hahn polynomials

Author: H. T. Koelink
Journal: Proc. Amer. Math. Soc. 124 (1996), 887-898
MSC (1991): Primary 33C45, 42A38
MathSciNet review: 1307541
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform, and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations for the Jacobi polynomials and the Parseval formula. In a special case this relation dates back to work by Bateman in 1933 and we follow a part of the historical development for these polynomials. Some applications of this relation are given.

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

  • 1. P. Appell and J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques. Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
  • 2. R. Askey, Continuous Hahn polynomials, J. Math. Phys. A: Math. Gen. 18 (1985), L1017--L1019. MR 87d:33021
  • 3. ------, Beta integrals and associated orthogonal polynomials, Number Theory (K. Alladi, ed.), Lecture Notes in Math., vol. 1395, Springer-Verlag, New York, 1989, pp. 84--121. MR 90k:33001
  • 4. R. Askey and J. Wilson, A set of hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 13 (1982), 651--655. MR 83h:33010
  • 5. ------, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc.,vol. 54, no. 319 (1985), Amer. Math. Soc., Providence, RI. MR 87a:05023
  • 6. N.M. Atakishiyev and S.K. Suslov, The Hahn and Meixner polynomials of imaginary argument and some of their applications, J. Math. Phys. A: Math. Gen. 18 (1985), 1583--1596. MR 87i:33021
  • 7. E. Badertscher and T.H. Koornwinder, Continuous Hahn polynomials of differential operator argument and analysis on Riemannian symmetric spaces of constant curvature, Canad. J. Math. 44 (1992), 750--773. MR 94k:43011
  • 8. W.N. Bailey, Generalized hypergeometric series, Cambridge Tracts 32, Cambridge University Press, London and New York, 1935; reprinted by Hafner Publishing Company, New York, 1972. MR 32:2625
  • 9. H. Bateman, Some properties of a certain set of polynomials, Tôhoku Math. J. 37 (1933), 23--38.
  • 10. ------, The polynomial $F_n(x)$, Ann. of Math. 35 (1934), 767--775.
  • 11. ------, Functions orthogonal in the Hermitean sense. A new application of basic numbers, Proc. Nat. Acad. Sci. U.S.A. 20 (1934), 63--66.
  • 12. ------, An orthogonal property of the hypergeometric polynomial, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 374--377. MR 4:83b
  • 13. F. Brafman, On Touchard polynomials, Canad. J. Math. 9 (1957), 191--193. MR 19:28a
  • 14. L. Carlitz, Some polynomials of Touchard connected with the Bernoulli numbers, Canad. J. Math. 9 (1957), 188--190. MR 19:27e
  • 15. ------, Bernoulli and Euler numbers and orthogonal polynomials, Duke Math. J. 26 (1959), 1--15. MR 21:2761
  • 16. T.S. Chihara, An introduction to orthogonal polynomials, Math. Appl., vol. 13, Gordon and Breach, New York, 1978. MR 58:1979
  • 17. A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables of integral transforms, Vol. 2, McGraw-Hill, New York, 1954. MR 16:468e
  • 18. G.H. Hardy, Notes on special systems of orthogonal functions (III): a system of orthogonal polynomials, Proc. Cambridge Philos. Soc. 36 (1940), 1--8;reprinted in Collected Papers, vol. IV (1969), Oxford University Press, London and New York, pp. 552--559. MR 1:141c
  • 19. E.G. Kalnins and W. Miller, $q$-Series and orthogonal polynomials associated with Barnes' first lemma, SIAM J. Math. Anal. 19 (1988), 1216--1231. MR 89m:33018
  • 20. R. Koekoek and R.F. Swarttouw, The Askey-scheme of orthogonal polynomials and its $q$-analogue, Report 94-05, Technical University Delft (1994).
  • 21. H.T. Koelink, Identities for $q$-ultraspherical polynomials and Jacobi functions, Proc. Amer. Math. Soc. 123 (1995), 2479--2487.
  • 22. T.H. Koornwinder, Special orthogonal polynomial systems mapped onto each other by the Fourier-Jacobi transfom, Polynômes Orthogonaux et Applications (C. Brezinski, A. Draux, A.P. Magnus, P. Maroni and A. Ronveaux, eds.), Lecture Notes in Math., vol. 1171, Springer-Verlag, 1985, pp. 174--183. MR 87g:33007
  • 23. ------, 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 in Math., vol. 1329, Springer-Verlag, 1988, pp. 46--72. MR 90b:33024
  • 24. ------, Meixner-Pollaczek polynomials and the Heisenberg algebra, J. Math. Phys. 30 (1989), 767--769. MR 90e:33037
  • 25. A.F. Nikiforov, S.K. Suslov and V.B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Ser. Comput. Phys., Springer-Verlag, New York and Berlin, 1991. MR 92m:33019
  • 26. S. Pasternack, A generalisation of the polynomial $F_n(x)$, London, Edinburgh, Dublin Philosophical Magazine and J. Science, Ser. 7 28 (1939), 209--226. MR 1:116a
  • 27. M. Rahman and S.K. Suslov, The Pearson integral and the beta integrals, SIAM J. Math. Anal. 25 (1994), 646--693. MR 95f:33001
  • 28. S.O. Rice, Some properties of ${}_3F_2(-n,n+1,\zeta ;1,p;v)$, Duke Math. J. 6 (1940), 108--119. MR 1:234b
  • 29. T.J. Stieltjes, Sur quelques intégrales définies et leur développement en fractions continues, Quart. J. Math., London 24 (1890), 370--382; reprinted in {\OE}uvres Complètes-Collected Papers, vol. II (G. van Dijk, ed.), Springer-Verlag, New York, 1993, pp. 382--395. MR 95g:01033
  • 30. ------, Recherches sur les fractions continues, Annales de la Faculté des Sciences de Toulouse 8 (1894), J.1--122, 9 (1895), A.1--47; reprinted in {\OE}uvres Complètes-Collected Papers, vol. II (G. van Dijk, ed.), Springer-Verlag, New York, 1993, pp. 406--570. MR 95g:01033
  • 31. G. Szeg\H{o}, Orthogonal polynomials, Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence, RI, 1975. MR 51:8724
  • 32. E.C. Titchmarsh, Introduction to the theory of Fourier integrals, 2nd ed., Oxford University Press, New York, 1948.
  • 33. J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canad. J. Math. 8 (1956), 305--320. MR 18:16f
  • 34. E.T. Whittaker and G.N. Watson, A course of modern analysis, 4th ed., Cambridge University Press, London and New York, 1927.
  • 35. J.A. Wilson, Some hypergeometric orthogonal polynomials, SIAM J. Math. Anal. 11 (1980), 690--701. MR 82a:33014
  • 36. M. Wyman and L. Moser, On some polynomials of Touchard, Canad. J. Math. 8 (1956), 321--322. MR 18:17a

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33C45, 42A38

Retrieve articles in all journals with MSC (1991): 33C45, 42A38

Additional Information

H. T. Koelink
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven (Heverlee), Belgium
Address at time of publication: Department of Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, the Netherlands

Keywords: Jacobi polynomials, continuous Hahn polynomials, Fourier transform
Received by editor(s): September 28, 1994
Additional Notes: Supported by a Fellowship of the Research Council of the Katholieke Universiteit Leuven.
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society