Symmetries and Special Functions

  • N. Ya. Vilenkin
  • A. U. Klimyk

Abstract

In the 18-th and 19-th centuries there appeared a great number of types of special functions to solve the equations of mathematical physics and to calculate the integrals. Many of them turned out to be special or limiting cases of the hypergeometric function F(α,β;γ;x), introduced in 1769 by L. Euler and scrutinized at the beginning of the 19-th century by Gauss. Gauss’ work triggered a flow of investigations which established different recurrent relations, differential equations, integral representations, generating functions, addition and multiplication theorems, asymptotic expansions for the hypergeometric function and its associates (Legendre, Gegenbauer, Hermite, Laguerre, Chebyshev polynomials; Bessel, Neumann, Macdonald, Whittaker functions, etc.), sought for relations between these functions, and calculated puzzling integrals involving them, etc. Books containing hundreds of pages were devoted to studies of some classes of special functions.

Keywords

Matrix Element Irreducible Representation Hypergeometric Function Spherical Function Jacobi Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    G. N. Ageev, N. Ja. Vilenkin, G. M. Javadov, A. I. Rubinshtein, “Multiplicative Systems of Functions and Harmonic Analysis on Nul-dimensional Groups,” ELM, Baku (1981) (in Russian).Google Scholar
  2. 2.
    A. U. Klimyk, “Matrix Elements and Clebsch-Gordon Coefficients of Group Representations,” Naukova Dumka, Kiev (1979) (in Russian).Google Scholar
  3. 3.
    A. U. Klimyk, A. M. Gavrilik, J. Math. Phys., 20: 1624 (1979).CrossRefGoogle Scholar
  4. 4.
    B. Gruber, A. U. Klimyk, J. Math. Phys., 22: 2762 (1981).CrossRefGoogle Scholar
  5. 5.
    A. U. Klimyk, B. Gruber, J. Math. Phys., 25: 743 (1984).CrossRefGoogle Scholar
  6. 6.
    A. U. Klimyk, B. Gruber, J. Math. Phys., 23: 1399 (1982).CrossRefGoogle Scholar
  7. 7.
    V. F. Molchanov, Mat. Sb., 99: 139 (1976).Google Scholar
  8. 8.
    N. Ja. Vilenkin, DAN SSSR, 113: 16, No. 1, (1957).Google Scholar
  9. 9.
    I. I. Kachurik, A. U. Klimyk, Reps. Math. Phys., 20: 333 (1984).CrossRefGoogle Scholar
  10. 10.
    A. V. Rosenbloom, L. V. Rosenbloom, Izv. AN BSSR, No. 4, 44 (1980)Google Scholar
  11. 11.
    N. Ja. Vilenkin, Mat. Sb., 68: 432 (1965).Google Scholar
  12. 12.
    M. S. Kildjushov, Soviet Nucl. Phys., 15: 197 (1972).CrossRefGoogle Scholar
  13. 13.
    S. K. Suslov, Soviet Nucl. Phys., 38: 1367 (1983).Google Scholar
  14. 14.
    N. Ya. Vilenkin, Trudy Mosk. Mat. Obsch., 12: 185 (1963).Google Scholar
  15. 15.
    I. M. Gel’fand, M. L. Zetlin, DAN SSSR, 71: 825 (1950).Google Scholar
  16. 16.
    I. M. Gel’fand, M. L. Zetlin, DAN SSSR, 71: 1017 (1950).Google Scholar
  17. 17.
    I. M. Gel’fand, M. I. Graev, Isv. AN SSSR, 29: 1329 (1965).Google Scholar
  18. 18.
    I. M. Gel’fand, M. I. Graev, N. Ja. Vilenkin, “Generalized Functions,” Vol. 5, Academic Press, New York (1966).Google Scholar
  19. 19.
    N. Ja. Vilenkin, Sb. Nauchn. Trudov Mosk. Ped. Inst., 39: 77 (1974).Google Scholar
  20. 20.
    F. A. Beresin, F. I. Karpelevich, DAN SSSR, 118: 9 (1958).Google Scholar
  21. 21.
    J. D. Louck, L. C. Biedenharn, J. Math. Anal. Appl., 59: 423 (1977).CrossRefGoogle Scholar
  22. 22.
    A. T. James, Ann. Math. Statistics, 25: 40 (1954).CrossRefGoogle Scholar
  23. 23.
    A. T. James, Ann. Math. Statistics, 39: 1711 (1968).Google Scholar
  24. 24.
    A. T. James, A. G. Constantine, Proc. London Math. Soc. (3), 29: 174 (1974).CrossRefGoogle Scholar
  25. 25.
    C. S. Herz, Ann. Math., 61: 474 (1955).CrossRefGoogle Scholar
  26. 26.
    N. Ja. Vilenkin, V. I. Paranuk, in “Some Problems of Mathematics and Physics,” Krasnodar (1969), p. 52.Google Scholar
  27. 27.
    H. Maass, J. Indian Math. Soc., 20: 117 (1956).Google Scholar
  28. 28.
    H. Maass, Math. Annalen, 135: 391 (1958).CrossRefGoogle Scholar
  29. 29.
    H. Maass, Math. Annalen, 137: 142 (1959).CrossRefGoogle Scholar
  30. 30.
    S. G. Gindikin, Uspechi Mat. Nauk, 19: 3 No. 4, (1964).Google Scholar
  31. 31.
    N. J. Vilenkin, L. M. Klesova, A. P. Pavliyk, in “Group-theoretical Methods in Physics,” Vol. 1, Nauka, Moscow (1980), p. 40.Google Scholar
  32. 32.
    D. Stanton, Amer. J. Math., 102: 625 (1980).CrossRefGoogle Scholar
  33. 33.
    D. Stanton, Geom. Dedicata, 10: 403 (1981).CrossRefGoogle Scholar
  34. 34.
    C. F. Duncl, Indiana Univ. Math. J., 25: 335 (1976).CrossRefGoogle Scholar
  35. 35.
    C. F. Duncl, Monats. Math., 85: 5 (1977).CrossRefGoogle Scholar
  36. 36.
    E. Bannai, T. Ito, Algebraic Combinatorics. I, (1984).Google Scholar
  37. 37.
    P. J. Feinsilver, Lect. Notes Math. 696 (1978).Google Scholar
  38. 38.
    G. Lions, M. Vergne, The Weil representations, Maslov index and theta series, Birkhauser, Basel, 1980.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • N. Ya. Vilenkin
    • 1
  • A. U. Klimyk
    • 2
  1. 1.Mathematical DepartmentThe Correspondence Pedagogical Institute (MGZPI)MoscowUSSR
  2. 2.Institute for Theoretical PhysicsKiev-130USSR

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