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Generalized Hermite Polynomials and the Bose-Like Oscillator Calculus

  • Marvin Rosenblum
Part of the Operator Theory: Advances and Applications book series (OT, volume 73)

Abstract

This paper studies a suitably normalized set of generalized Hermite polynomials and sets down a relevant Mehler formula, Rodrigues formula, and generalized translation operator. Weighted generalized Hermite polynomials are the eigenfunctions of a generalized Fourier transform which satisfies an F. and M. Riesz theorem on the absolute continuity of analytic measures. The Bose-like oscillator calculus, which generalizes the calculus associated with the quantum mechanical simple harmonic oscillator, is studied in terms of these polynomials.

Keywords

Entire Function Selfadjoint Operator Confluent Hypergeometric Function Riesz Theorem Rodrigues Formula 
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.
    Ahiezer and Glazman, Theory of Linear Operators in Hilbert Space , Vol 1 ,Frederick Ungar, New York, 1961.Google Scholar
  2. 2.
    R. Askey, Orthogonal Polynomials and Special Functions ,Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1975.CrossRefGoogle Scholar
  3. 3.
    L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics ,Encyclopedia of Mathematics and its Applications, Vol 9, Addison-Wesley, Reading, Massachusetts, 1981.Google Scholar
  4. 4.
    T. S. Chihara, Generalized Hermite Polynomials ,Thesis, Purdue, 1955.Google Scholar
  5. 5.
    T. S. Chihara, An Introduction to Orthogonal Polynomials ,Gordon and Breach, New York, London, Paris, 1984.Google Scholar
  6. 6.
    F. M. Cholewinski and D. T. Haimo, Classical analysis and the generalized heat equation ,SIAM Review 10 (1968), 67–80.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. M. Cholewinski, Generalized Foch spaces and associated operators ,SIAM J. Math. Analysis 15 (1984), 177–202.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    F.M. Cholewinski, The Finite Calculus Associated with Bessel Functions ,Contemporary Mathematics Vol. 75, American Mathematics Society, Providence, Rhode Island, 1988.Google Scholar
  9. 9.
    H.S.M. Coxeter, Introduction to Geometry ,John Wiley, N. Y., London, Sydney, Toronto, 1969.zbMATHGoogle Scholar
  10. 10.
    D.J.Dickinson and S.A. Warsi, On a generalized Hermite polynomial and a problem of Carlitz ,Boll. Un. Mat. Ital. (3) 18 (1963), 256–259.MathSciNetzbMATHGoogle Scholar
  11. 11.
    C. F. Dunkl, Integral kernels with reflection group invariance ,Canadian J. Math. 43 (1991), 1213–1227.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    C. F. Dunkl, Hankel transforms associated to finite reflection groups ,Contemporary Math. to appear.Google Scholar
  13. 13.
    M. Dutta, S.K. Chatterjea and K. L. More, On a class of generalized Hermite polynomials ,Bull. of the Inst. of Math. Acad. Sinica 3 (1975), 377–381.MathSciNetzbMATHGoogle Scholar
  14. 14.
    H. Dym and H. P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem ,Vol 31 , Probability and Mathematical Statistics, Academic Press, New York, San Francisco, London, 1976.Google Scholar
  15. 15.
    A. Erdélyi, Higher Transcendental Functions, Vol 1, 2 ,3 ,McGraw-Hill, New York, 1980.Google Scholar
  16. 16.
    A. Erdélyi, Tables of Integral Transforms, Vol 1, 2 ,McGraw-Hill, New York, 1954.Google Scholar
  17. 17.
    J. Glimm and A. Jaffe, Quantum Physics ,Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
  18. 18.
    E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups ,Amer. Math. Soc. Colloquium Publ. Vol. 31, American Mathematics Society, Providence, Rhode Island, 1957.Google Scholar
  19. 19.
    N. N. Lebedev, Special Functions and their Applications ,Translated by R. A. Silverman, Dover, New York, 1972.zbMATHGoogle Scholar
  20. 20.
    J. D. Louck, Extension of the Kibble-Slepian formula to Hermite polynomials using Boson operator methods ,Advances in Applied Math. 2 (1981), 239–249.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions ,Trans. Amer. Math. Soc. 118 (1965), 17–92.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    N. Mukunda, E.C.G. Sudershan, J.K. Sharma, and C.L. Mehta, Representations and properties of para-B ose oscillator operators. I. Energy position and momentum eigenstates ,J. Math. Phys. 21 (1980), 2386–2394.MathSciNetCrossRefGoogle Scholar
  23. 23.
    E. Nelson, Analytic vectors ,Annals of Math. 70 (1959), 572–615.CrossRefzbMATHGoogle Scholar
  24. 24.
    Y. Ohnuki and S.Kamefuchi, Quantum Field Theory and Parastatistics ,University of Tokyo Press, Springer-Verlag Berlin Heidelberg New York, 1982.CrossRefzbMATHGoogle Scholar
  25. 25.
    E. D. Rainville, Special Functions ,Chelsea, Bronx, New York, 1971.zbMATHGoogle Scholar
  26. 26.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics ,I, Functional Analysis, Academic Press, San Diego New York Berkeley, 1980.zbMATHGoogle Scholar
  27. 27.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics ,II Fourier Analysis, Self-Adjointness, Academic Press, New York San Francisco London, 1975.Google Scholar
  28. 28.
    L. J. Slater, Confluent Hypergeometric Functions ,Cambridge University Press, London and New York, 1960.zbMATHGoogle Scholar
  29. 29.
    G. Szego, Orthogonal Polynomials ,Amer. Math. Soc. Colloquium Publ. Vol. 23, American Mathematics Society, New York, 1939.Google Scholar
  30. 30.
    G. N. Watson, A Treatise on the Theory of Bessel Functions,2nd Edition ,Cambridge University Press, Cambridge, Great Britain, 1966.zbMATHGoogle Scholar
  31. 31.
    E. P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations? ,Phys. Rev. 77 (1950), 711–712.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Marvin Rosenblum
    • 1
  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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