Skip to main content
SpringerLink
Log in

Symmetries and Special Functions

  • Chapter
Symmetries in Science II

  • 148 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. A. U. Klimyk, “Matrix Elements and Clebsch-Gordon Coefficients of Group Representations,” Naukova Dumka, Kiev (1979) (in Russian).

    Google Scholar 

  3. A. U. Klimyk, A. M. Gavrilik, J. Math. Phys., 20: 1624 (1979).

    Article  Google Scholar 

  4. B. Gruber, A. U. Klimyk, J. Math. Phys., 22: 2762 (1981).

    Article  CAS  Google Scholar 

  5. A. U. Klimyk, B. Gruber, J. Math. Phys., 25: 743 (1984).

    Article  Google Scholar 

  6. A. U. Klimyk, B. Gruber, J. Math. Phys., 23: 1399 (1982).

    Article  Google Scholar 

  7. V. F. Molchanov, Mat. Sb., 99: 139 (1976).

    Google Scholar 

  8. N. Ja. Vilenkin, DAN SSSR, 113: 16, No. 1, (1957).

    Google Scholar 

  9. I. I. Kachurik, A. U. Klimyk, Reps. Math. Phys., 20: 333 (1984).

    Article  Google Scholar 

  10. A. V. Rosenbloom, L. V. Rosenbloom, Izv. AN BSSR, No. 4, 44 (1980)

    Google Scholar 

  11. N. Ja. Vilenkin, Mat. Sb., 68: 432 (1965).

    Google Scholar 

  12. M. S. Kildjushov, Soviet Nucl. Phys., 15: 197 (1972).

    Article  Google Scholar 

  13. S. K. Suslov, Soviet Nucl. Phys., 38: 1367 (1983).

    Google Scholar 

  14. N. Ya. Vilenkin, Trudy Mosk. Mat. Obsch., 12: 185 (1963).

    Google Scholar 

  15. I. M. Gel’fand, M. L. Zetlin, DAN SSSR, 71: 825 (1950).

    Google Scholar 

  16. I. M. Gel’fand, M. L. Zetlin, DAN SSSR, 71: 1017 (1950).

    Google Scholar 

  17. I. M. Gel’fand, M. I. Graev, Isv. AN SSSR, 29: 1329 (1965).

    Google Scholar 

  18. I. M. Gel’fand, M. I. Graev, N. Ja. Vilenkin, “Generalized Functions,” Vol. 5, Academic Press, New York (1966).

    Google Scholar 

  19. N. Ja. Vilenkin, Sb. Nauchn. Trudov Mosk. Ped. Inst., 39: 77 (1974).

    Google Scholar 

  20. F. A. Beresin, F. I. Karpelevich, DAN SSSR, 118: 9 (1958).

    Google Scholar 

  21. J. D. Louck, L. C. Biedenharn, J. Math. Anal. Appl., 59: 423 (1977).

    Article  Google Scholar 

  22. A. T. James, Ann. Math. Statistics, 25: 40 (1954).

    Article  Google Scholar 

  23. A. T. James, Ann. Math. Statistics, 39: 1711 (1968).

    Google Scholar 

  24. A. T. James, A. G. Constantine, Proc. London Math. Soc. (3), 29: 174 (1974).

    Article  CAS  Google Scholar 

  25. C. S. Herz, Ann. Math., 61: 474 (1955).

    Article  Google Scholar 

  26. N. Ja. Vilenkin, V. I. Paranuk, in “Some Problems of Mathematics and Physics,” Krasnodar (1969), p. 52.

    Google Scholar 

  27. H. Maass, J. Indian Math. Soc., 20: 117 (1956).

    Google Scholar 

  28. H. Maass, Math. Annalen, 135: 391 (1958).

    Article  Google Scholar 

  29. H. Maass, Math. Annalen, 137: 142 (1959).

    Article  Google Scholar 

  30. S. G. Gindikin, Uspechi Mat. Nauk, 19: 3 No. 4, (1964).

    Google Scholar 

  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. D. Stanton, Amer. J. Math., 102: 625 (1980).

    Article  Google Scholar 

  33. D. Stanton, Geom. Dedicata, 10: 403 (1981).

    Article  Google Scholar 

  34. C. F. Duncl, Indiana Univ. Math. J., 25: 335 (1976).

    Article  Google Scholar 

  35. C. F. Duncl, Monats. Math., 85: 5 (1977).

    Article  Google Scholar 

  36. E. Bannai, T. Ito, Algebraic Combinatorics. I, (1984).

    Google Scholar 

  37. P. J. Feinsilver, Lect. Notes Math. 696 (1978).

    Google Scholar 

  38. G. Lions, M. Vergne, The Weil representations, Maslov index and theta series, Birkhauser, Basel, 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Department, The Correspondence Pedagogical Institute (MGZPI), Moscow, 109004, USSR

    N. Ya. Vilenkin

  2. Institute for Theoretical Physics, Kiev-130, 252130, USSR

    A. U. Klimyk

Authors
  1. N. Ya. Vilenkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. U. Klimyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Southern Illinois University, Carbondale, Illinois, USA

    Bruno Gruber  & Romuald Lenczewski  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer Science+Business Media New York

About this chapter

Cite this chapter

Vilenkin, N.Y., Klimyk, A.U. (1986). Symmetries and Special Functions. In: Gruber, B., Lenczewski, R. (eds) Symmetries in Science II. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1472-2_47

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-1472-2_47

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-1474-6

  • Online ISBN: 978-1-4757-1472-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation