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Continuous Hahn Functions as Clebsch-Gordan Coefficients

  • Wolter Groenevelt
  • Erik Koelink
  • Hjalmar Rosengren
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 13)

Abstract

An explicit bilinear generating function for Meixner-Pollaczek polynomials is proved. This formula involves continuous dual Hahn polynomials, Meixner-Pollaczek functions, and non-polynomial 3 F 2-hypergeometric functions that we consider as continuous Hahn functions. An integral transform pair with continuous Hahn functions as kernels is also proved. These results have an interpretation for the tensor product decomposition of a positive and a negative discrete series representation of su(1, 1) with respect to hyperbolic bases, where the Clebsch-Gordan coefficients are continuous Hahn functions.

Keywords

Tensor Product Orthogonal Polynomial Summation Formula Discrete Series Generalize Eigenvector 
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References

  1. Andrews, G. E., Askey, R., and Roy, R. (1999). Special Functions, volume 71 of Encycl. Math. Appl. Cambridge Univ. Press, Cambridge.zbMATHGoogle Scholar
  2. Bailey, W. N. (1972). Generalized Hypergeometric Series. Hafner, New York.Google Scholar
  3. Basu, D. and Wolf, K. B. (1983). The Clebsch-Gordan coefficients of the three-dimensianal Lorentz algebra in the parabolic basis. J. Math. Phys., 24:478–500.CrossRefMathSciNetzbMATHGoogle Scholar
  4. Braaksma, B. L. J. and Meulenbeld, B. (1967). Integral transforms with generalized Legendre functions as kernels. Compositio Math., 18:235–287.MathSciNetzbMATHGoogle Scholar
  5. de Jeu, M. (2003). Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights. Ann. Prob., 31:1205–1227. Available electronically: math.CA/0111019.CrossRefzbMATHGoogle Scholar
  6. Engliš, M., Hille, S. C., Peetre, J., Rosengren, H., and Zhang, G. (2000). A new kind of Hankel-Toeplitz type operator connected with the complementary series. Arab J. Math. Sci., 6:49–80.MathSciNetzbMATHGoogle Scholar
  7. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). Higher Transcendental Functions, volume I, II. McGraw-Hill, New York-Toronto-London.Google Scholar
  8. Götze, F. (1965). Verallgemeinerung einer integraltransformation von Mehler-Fock durch den von Kuipers und Meulenbeld eingefürten Kern p m,nk(z). Indag. Math., 27:396–404.Google Scholar
  9. Granovskii, Y. I. and Zhedanov, A. S. (1993). New construction of 3nj-symbols. J. Phys. A: Math. Gen., 26:4339–4344.CrossRefMathSciNetGoogle Scholar
  10. Groenevelt, W. (2003). Laguerre functions and representations of su(1, 1). Indag. Math. To appear.Google Scholar
  11. Groenevelt, W. and Koelink, E. (2002). Meixner functions and polynomials related to Lie algebra representations. J. Phys. A: Math. Gen., 35:65–85.CrossRefMathSciNetzbMATHGoogle Scholar
  12. Ismail, M. E. H. and Stanton, D. (2002). q-integral and moment representations for q-orthogonal polynomials. Canad. J. Math., 54:709–735.MathSciNetzbMATHGoogle Scholar
  13. Koekoek, R. and Swarttouw, R. F. (1998). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Technical Report 98-17, Technical University Delft, Delft.Google Scholar
  14. Koelink, E. Spectral theory and special functions. In Marcelldn, F., Van Assche, W., and Alvarez-Nordarse, R., editors, Proceedings of the 2000 SIAG OP-SF Summer School on Orthogonal Polynomials and Special Functions (Laredo, Spain). Nova Science Publishers, Hauppauge, NY. Available electronically: math.CA/0107036.Google Scholar
  15. Koelink, H. T. and Van der Jeugt, J. (1998). Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal., 29:794–822.CrossRefMathSciNetzbMATHGoogle Scholar
  16. Koornwinder, T. H. (1984). Jacobi functions and analysis on noncompact semisimple Lie groups. In Askey, R. A., Koornwinder, T. H., and Schempp, W., editors, Special functions: Group Theoretical Aspects and Applications, Math. Appl., pages 1–85. D. Reidel Publ. Comp., Dordrecht.Google Scholar
  17. Koornwinder, T. H. (1988). Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials. In Alfaro, et al., M., editor, Orthogonal Polynomials and Their Applications, volume 1329 of Lecture Notes Math., pages 46–72. Springer-Verlag, Berlin.Google Scholar
  18. Masson, D. R. and Repka, J. (1991). Spectral theory of Jacobi matrices in l2(Z) and the su(1, 1) Lie algebra. SIAM J. Math. Anal., 22:1131–1146.CrossRefMathSciNetzbMATHGoogle Scholar
  19. Mukunda, N. and Radhakrishnan, B. (1974). Clebsch-Gordan problem and coefficients for the three-dimensional Lorentz group in a continuous basis. II. J. Math. Phys., 15:1332–1342.CrossRefGoogle Scholar
  20. Neretin, Y. A. (1986). Discrete occurrences of representations of the complementary series in tensor products of unitary representations. Funktsional. Anal. i Prilozhen., 20:79–80. English translation: Funct. Anal. Appl. 20 (1986), 68–70.zbMATHMathSciNetCrossRefGoogle Scholar
  21. Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.Google Scholar
  22. Paris, R. B. and Kaminski, D. (2001). Asymptotics and Mellin-Barnes Integrals, volume 85 of Encycl. Math. Appl. Cambridge Univ. Press, Cambridge.zbMATHGoogle Scholar
  23. Schmüdgen, K. (1990). Unbounded Operator Algebras and Representation Theory, volume 37 of Operator theory: Advances and Applications. Birkhüuser Verlag, Basel.Google Scholar
  24. Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge Univ. Press, Cambridge.zbMATHGoogle Scholar
  25. Slater, L. J. (1966). Generalized Hypergeometric Functions. Cambridge Univ. Press, Cambridge.zbMATHGoogle Scholar
  26. Van der Jeugt, J. (1997). Coupling coefficients for Lie algebra representations and addition formulas for special functions. J. Math. Phys., 38:2728–2740.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Vilenkin, N. J. and Klimyk, A. U. (1991). Representations of Lie Groups and Special Functions, volume 1. Kluwer Academic Publishers, Dordrecht.Google Scholar
  28. Whittaker, E. T. and Watson, G. N. (1963). A Course of Modern Analysis. Cambridge Univ. Press, Cambridge, fourth edition.zbMATHGoogle Scholar
  29. Wilson, J. A. (1980). Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal., 11:690–701CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Wolter Groenevelt
    • 1
  • Erik Koelink
    • 1
  • Hjalmar Rosengren
    • 2
  1. 1.Technische Universiteit Delft EWI-TWADelftThe Netherlands
  2. 2.Department of MathematicsChalmers University of Technology and Göteborg UniversityGöteborgSweden

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