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Wilson Function Transforms Related to Racah Coefficients

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Abstract

The irreducible \(*\)-representations of the Lie algebra \({\mathfrak {\rm u}}(1,1)\)consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch–Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for \({\mathcal U}_q({\mathfrak{\rm u}}(1,1))\), which turn out to be Askey–Wilson functions and Askey–Wilson polynomials.

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References

  1. Akhiezer, N.I.: The classical moment problem and some related questions in analysis. Hafner, New York (1965)

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encycl. Math. Appl. vol. 71. Cambridge University Press, Cambridge (1999)

  3. Askey, R., Wilson, J.: A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols. SIAM J. Math. Anal. 10, 1008–1016 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 54(319), 1–55 (1985)

    MathSciNet  Google Scholar 

  5. Bargmann, V.: Irreducible unitary representations of the Lorentz group. Ann. Math. 48, 568–640 (1947)

    Article  MathSciNet  Google Scholar 

  6. Biedenharn, L.C.: An identity satisfied by Racah coefficients. J. Math. Phys. 31, 287–293 (1953)

    MathSciNet  MATH  Google Scholar 

  7. Biedenharn, L.C., Louck, J.D.: Angular momentum in quantum physics. Theory and Application. Addison-Wesley, Reading, Massachusetts (1981)

    MATH  Google Scholar 

  8. Buffenoir, E., Roche, Ph.: Tensor product of principal unitary representations of quantum Lorentz group and Askey-Wilson polynomials. J. Math. Phys. 41, 7715–7751 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burban, I.M., Klimyk, A.U.: Representations of the quantum algebra \(U_q({\mathfrak{s}}{u}_{1,1})\). J. Phys., A, Math. Gen. 26, 2139–2151 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Christiansen, J.S.: Inderminate Moment Problems within the Askey-scheme. PhD thesis, University of Copenhagen. http://www.math.ku.dk/ stordal/thesis.pdf(2004)

  11. Elliott, J.P.: Theoretical studies in nuclear structure V: The matrix elements of non-central forces with an application to the \(2p\)-shell. Proc. Roy. Soc. A218, 345–370 (1953)

    Google Scholar 

  12. Gasper, G., Rahman, M.: Basic Hypergeometric series, 2nd ed. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  13. Groenevelt, W.: The Wilson function transform. Int. Math. Res. Not. 52, 2779–2817 (2003)

    Article  MathSciNet  Google Scholar 

  14. Groenevelt, W.: Bilinear summation formulas from quantum algebra representations. Ramanujan J. 8, 383–416 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Groenevelt, W., Koelink, E.: Meixner functions and polynomials related to Lie algebra representations. J. Phys., A, Math. Gen. 35, 65–85 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gupta, D.P., Ismail, M.E.H., Masson, D.R.: Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual \(q\)-Hahn polynomials. J. Comput. Appl. Math. 68, 115–149 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kalnins, E.G., Miller Jr., W.: A note on tensor products of \(q\)-algebra representations and orthogonal polynomials. J. Comp. Appl. Math. 68, 197–207 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kerimov, G.A., Verdiyev, Y.A.: Clebsch–Gordan problem for three-dimensional Lorentz group in the elliptic basis. I. Tensor product of continuous series. J. Phys., A, Math. Gen. 31, 3573–3593 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kerimov, G.A., Verdiyev, Y.A.: Clebsch–Gordan problem for the three-dimensional Lorentz group in the elliptic basis: II. Tensor products involving discrete series. J. Phys., A, Math. Gen. 32, 3385–3395 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kirillov, A.N., Reshetikhin, N.Yu.: Representations of the algebra \({\mathcal U}_q(\mathfrak{sl}(2))\), \(q\)-orthogonal polynomials and invariants of links, In: Kac, V.G. (ed.) Infinite-dimensional Lie Algebras and Groups. Adv. Ser. Math. Phys., vol. 7, pp. 285–339. World Sci., Teaneck, New Jersey (1989)

    Google Scholar 

  21. Koekoek, R., Swarttouw, R.F.: The Askey-Scheme of Hypergeometric orthogonal polynomials and its q-analogue. Report 98-17. Technical University Delft, Delft (1998)

    Google Scholar 

  22. Koelink, H.T.: Askey–Wilson polynomials and the quantum \({\rm SU}(2)\) group: Survey and applications. Acta Appl. Math. 44, 295–352 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Koelink, H.T., Koornwinder, T.H.: The Clebsch–Gordan coefficients for the quantum group \(S_\mu U(2)\) and \(q\)-Hahn polynomials. Indag. Math. 51, 443–456 (1989)

    MathSciNet  Google Scholar 

  24. Koelink, E., Stokman, J.V.: The Askey–Wilson function transform. Int. Math. Res. Not. 22, 1203–1227 (2001)

    Article  MathSciNet  Google Scholar 

  25. Koelink, E., Stokman, J.V. with an appendix by M. Rahman: Fourier transforms on the quantum \(SU(1,1)\) group. Publ. Res. Inst. Math. Sci., Kyoto Univ. 37, 621–715 (2001)

  26. Koelink, H.T., Van der Jeugt, J.: Convolutions for orthogonal polynomials from Lie and quantum algebra representations. SIAM J. Math. Anal. 29, 794–822 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Koelink, H.T., Van der Jeugt, J.: Bilinear generating functions for orthogonal polynomials. Constr. Approx. 14, 481–497 (1999)

    Article  Google Scholar 

  28. Koornwinder, T.H.: Clebsch–Gordan coefficients for \(SU(2)\) and Hahn polynomials. Nieuw Arch. Wiskd. 29, 140–155 (1981)

    MathSciNet  MATH  Google Scholar 

  29. Koornwinder, T.H.: A group theoretic interpretation of Wilson polynomials, In: Complex Analysis and Applications '85 (Varna, 1985), pp. 335–355. Publ. House Bulgar. Acad. Sci., Sofia (1986)

    Google Scholar 

  30. Koornwinder, T.H.: Askey–Wilson polynomials as zonal spherical functions on the \(SU(2)\) quantum group. SIAM J. Math. Anal. 24, 795–813 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lievens, S., Van der Jeugt, J.: \({3nj}\)-coefficients of \({{\mathfrak {s}}{\rm {u}}}(1,1)\) as connection coefficients between orthogonal polynomials in \(2\) variables. J. Math. Phys. 42, 3824–3849 (2002)

    Article  Google Scholar 

  32. Noumi, M., Mimachi, K.: Askey–Wilson polynomials as spherical functions on \({\rm SU}\sb q(2)\), In: Quantum groups (Leningrad, 1990), Lecture Notes in Math., vol. 1510, pp. 98–103. Springer, Berlin Heidelberg New York (1992)

    Google Scholar 

  33. Racah, G.: Theory of complex spectra I, II, III, IV. Phys. Rev. 61, 186–197 (1942), 62, 438–462 (1942), 63, 367–382 (1943), 76, 1352–1365 (1949)

    Article  Google Scholar 

  34. Rosengren, H.: A new quantum algebraic interpretation of the Askey–Wilson polynomials. Contemp. Math. 254, 371–394 (2000)

    MathSciNet  Google Scholar 

  35. Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Operator Theory, vol. 37. Birkhäuser, Basel (1990)

    Google Scholar 

  36. Smirnov, Y.F., Suslov, S.K., Shirokov, A.M.: Clebsch–Gordan coefficients and Racah coefficients for the \(SU(2)\) and \(SU(1,1)\) groups as the discrete analogues of the Pöschl–Teller potential wavefunctions. J. Phys., A, Math. Gen. 17, 2157–2175 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  37. Stokman, J.V.: An expansion formula for the Askey–Wilson function. J. Approx. Theory 114, 308–342 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Stokman, J.V.: Difference Fourier transforms for nonreduced root systems. Sel. Math., New Ser. 9, 409–494 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vaksman, L.L.: \(q\)-analogues of Clebsch–Gordan coefficients, and the algebra of functions on the quantum group \(SU(2)\). Dokl. Akad. Nauk SSSR 306, 269–271 (1989)

    MathSciNet  Google Scholar 

  40. Van der Jeugt, J.: \(3nj\)-coefficients and orthogonal polynomials of hypergeometric type. In: Koelink, E., Van Assche W. (eds.) Orthogonal Polynomials and Special Functions. Lecture Notes in Math. vol. 1817, pp. 25–92. Springer, Berlin Heidelberg New York (2003)

  41. Vilenkin, N.J., Klimyk, A.U.: Representations of Lie Groups and Special Functions, vol. 1 Kluwer, Dordrecht (1991)

    Google Scholar 

  42. Wilson, J.A.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands

    Wolter Groenevelt

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  1. Wolter Groenevelt
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Correspondence to Wolter Groenevelt.

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This research was done during my stay at the Department of Mathematics at Chalmers University of Technology and Göteborg University in Sweden, supported by a NWO-TALENT stipendium of the Netherlands Organization for Scientific Research (NWO).

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Groenevelt, W. Wilson Function Transforms Related to Racah Coefficients. Acta Appl Math 91, 133–191 (2006). https://doi.org/10.1007/s10440-006-9024-7

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