Skip to main content
SpringerLink
Log in

Representations of Lie Groups and Special Functions

  • Chapter
Representation Theory and Noncommutative Harmonic Analysis II

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 59))

Abstract

At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the exponential function, the probability integral and so on. Elliptic integrals proved to be the most important. They are connected with rectification of arcs of certain curves. The remarkable idea of Abel to replace these integrals by the corresponding inverse functions led to the creation of the theory of elliptic 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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

  • Alain, P.N.D. (1977) On the representations of the groups SO(3), 0(2,1) and M(2) defined in terms of Lamé, Mathieu and parabolic functions. Rep. Math. Phys. 11, No. 1, 17–36.Zbl. 357.22011

    MATH  Google Scholar 

  • Alishauskas, S.I. (1983) Relationship between spaces of representations and the Wigner—Racah calculus for compact groups. Fizika elementarnykh chastits i atomnogo yadra 14, 1336–1379.

    Google Scholar 

  • Alishauskas, S.I. (1983) Relationship between spaces of representations and the Wigner—Racah calculus for compact groups. Fizika elementarnykh chastits i atomnogo yadra English transl.: Sov. J. Part. Nucl. 14, 563–597 (1983)

    Google Scholar 

  • Andrews, G.E., and Askey, R. (1985) Classical orthogonal polynomials Lect. Notes Math. 1171, 36–62.Zbl. 298.33016

    MathSciNet  Google Scholar 

  • Andrews, M., and Gunson, J. (1964) Complex angular momenta and many-particles states. I. J. Math. Phys. 5, 1391–1401

    MathSciNet  Google Scholar 

  • Askey, R. (1975a) Orthogonal Polynomials and Special Functions. Philadelphia: SIAM, 110 pp.Zbl. 298.33008

    Google Scholar 

  • Askey, R. (1975b) (Ed) Theory and Applications of Special Functions. New York: Academic Press, XI, 560 pp.Zbl 316.00014

    Google Scholar 

  • Askey, R., and Gasper, G. (1977) Convolution structures for Laguerre polynomials. J. Anal Math. 31, 48–68.Zbl. 347.33006

    MathSciNet  MATH  Google Scholar 

  • Askey, R., Koornwinder, T.H., and Schempp, W. (Eds) (1984) Special Functions: Group Theoretical Aspects and Applications. Dordrecht: Reidel, XXXIV, 311 pp.Zbl. 543.00007

    Google Scholar 

  • Askey, R., and Wilson, J. (1979) A set of orthogonal polynomials that generalize the Racah coefficients or 6j-symbols. SIAM J. Math. Anal. 10, 1008–1016.Zbl. 437.33014

    MathSciNet  MATH  Google Scholar 

  • Auslander, L., and Moore, C.C. (1966) Unitary Representations of Solvable Lie Groups. Providence: Am. Math. Soc., 199 pp.Zbl. 204, 142

    Google Scholar 

  • Bailey, W.N. (1935) Generalized Hypergeometric Series. Cambridge: Camb. Univ. Press, VII, 108 pp.Zbl. 11, 23

    Google Scholar 

  • Bannai, E., and Ito, T. (1984) Algebraic Combinatorics. I. California: Benjamin, 425 pp.Zbl. 555.05019

    MATH  Google Scholar 

  • Bargmann, V. (1947) Irreducible unitary representations of the Lorentz group. Ann. Math., II. Ser. 48, 568–640.Zbl. 45, 388

    MathSciNet  MATH  Google Scholar 

  • Basu, D., and Wolf, K.B. (1982) The unitary irreducible representations of SL(2,R) in all subgroup reductions. J. Math. Phys. 23, 189–205.Zbl. 512.22015

    MathSciNet  MATH  Google Scholar 

  • Basu, D., and Wolf, K.B. (1983) The Clebsch—Gordan coefficients of the three-dimensional Lorentz algebra in the parabolic basis. J. Math. Phys. 24, 478–500.Zbl. 513.22014

    MathSciNet  MATH  Google Scholar 

  • Berezin, F.A., and Karpelevich, F.I. (1958) Zonal spherical functions and Laplace operators on certain symmetric spaces. Dokl. Akad. Nauk SSSR 118, No. 1, 9–12(Russian) Zbl. 78, 92

    MATH  Google Scholar 

  • Biedenharn, L.C., and Louck, J.D. (1981a) Angular Momentum in Quantum Physics. Massachusetts: Addison—Wesley, 716 pp.Zbl. 474.00023

    MATH  Google Scholar 

  • Biedenharn, L.C., and Louck, J.D. (1981b) The Racah—Wigner Algebra in Quantum Theory. Massachusetts: Addison—Wesley, 534 pp.Zbl. 474.00024

    MATH  Google Scholar 

  • Braaksma, B.L., and Meulenbeld, B. (1967) Integral transforms with generalized Legendre functions as kernels. Compos. Math. 18, 235–287.Zbl. 177, 399

    MathSciNet  MATH  Google Scholar 

  • Clerc, J.L. (1987) Le comportement à l’infini des fonctions de Bessel généralisées. Adv. Math. 66, 31–61.Zbl. 629.43009

    MathSciNet  MATH  Google Scholar 

  • Dunkl, C.F. (1976) A Krawtchouk polynomial addition theorem and wreath products of symmetric groups. Indiana Univ. Math. J. 25, 335–358.Zbl. 326.33008

    MathSciNet  MATH  Google Scholar 

  • Dunkl, C.F. (1978) An addition theorem for some q-Hahn polynomials. Monatsh. Math. 85, 5–37.Zbl. 368.33005

    MathSciNet  MATH  Google Scholar 

  • Dunkl, C.F. (1979) Orthogonal functions on some permutation groups. Proc. Symp. Pure Math. 34, 129–147.Zbl. 404.20005

    MathSciNet  Google Scholar 

  • Dunkl, C.F. (1981) Cube group invariant spherical harmonics and Krawtchouk polynomials. Math. Z. 177, 561–577.Zbl. 451.33005

    MathSciNet  MATH  Google Scholar 

  • Durand, L. (1979) Addition formulas for Jacobi, Gegenbauer, Laguerre and hyperbolic Bessel functions of the second kind. SIAM J. Math. Anal 10, 425–437.Zbl. 409.33008

    MathSciNet  MATH  Google Scholar 

  • Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricorni, F. (1953) Higher Transcendental Functions. I. New York: McGraw—Hill, XXVI 302 pp.Zbl. 51, 303

    Google Scholar 

  • Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricorni, F. (1955) Higher Transcendental Functions. II. New York: McGraw—Hill, XVII, 396 pp.Zbl. 52, 295

    Google Scholar 

  • Feinsilver, P.J. (1978) Special functions, probability semigroups and Hamiltonian flows. Lect. Notes Math. 696, 1–112.Zbl 394.33002

    MathSciNet  Google Scholar 

  • Flensted-Jensen, M. (1972) Paley—Wiener type theorems for a differential operator connected with symmetric spaces. Ark. Math. 10, 143–162.Zbl. 233.42012

    MathSciNet  MATH  Google Scholar 

  • Gasper, G. (1971) Positivity and the convolution structure for Jacobi series. Ann. Math., II. Ser. 93, 112–118.Zbl. 208, 81

    MathSciNet  MATH  Google Scholar 

  • Gelbart, S. (1972) Harmonics on Stiefel manifolds and generalized Hankel transforms. Bull. Am. Math. Soc. 78, 451–455.Zbl. 235.43012

    MathSciNet  MATH  Google Scholar 

  • Gelbart, S. (1974) A theory of Stiefel harmonics. Trans. Am. Math. Soc. 192, 29–50.Zbl. 292.22016

    MathSciNet  MATH  Google Scholar 

  • Gel’fand, I.M. (1950) Spherical functions on symmetric Riemannian spaces. Dokl. Akad. Nauk SSSR 70, No. 1, 5–8.Zbl. 38, 274.

    MATH  Google Scholar 

  • Gel’fand, I.M. (1950) English transi.: Transl., II. Ser., Am. Math. Soc. 37,39–43 (1964)

    Google Scholar 

  • Gel’fand, I.M. (1986) General theory of hypergeometric functions. Dokl. Akad. Nauk SSSR 288, No. 1, 14–18.

    MathSciNet  Google Scholar 

  • Gel’fand, I.M. (1986) General theory of hypergeometric functions. English transl.: Sov. Math., Dokl. 33, 573–577 (1987).Zbl. 645.33010

    Google Scholar 

  • Gel’fand, I.M., and Berezin, F.A. (1956) Several remarks on the theory of spherical functions on symmetric Riemannian manifolds. Tr. Mosk. Mat. 0.-va. 3, 311–356.Zbl. 75, 118

    MathSciNet  Google Scholar 

  • Gel’fand, I.M., and Gel’fand, S.I. (1986) Generalized hypergeometric equations. Dokl. Akad. Nauk SSSR 288, No. 2, 279–283.

    MathSciNet  Google Scholar 

  • Gel’fand, I.M., and Gel’fand, S.I. (1986) Generalized hypergeometric equations. English transl.: Sov. Math., Dokl. 33, 643–646 (1987).Zbl. 634.58030

    Google Scholar 

  • Gel’fand, I.M., and Graev, M.I. (1963) Representations of the group of matrices of the second order with elements from a locally compact field and special functions on locally compact fields. Usp. Mat. Nauk 18, 29–99.Zbl. 166, 402

    MathSciNet  MATH  Google Scholar 

  • Gel’fand, I.M., and Graev, M.I. (1963) Representations of the group of matrices of the second order with elements from a locally compact field and special functions on locally compact fields. English transl.: Russ. Math. Surv. 18, No.4, 29–100 (1963)

    MATH  Google Scholar 

  • Gel’fand, I.M., and Graev, M.I. (1965) Finite-dimensional irreducible representations of the unitary and full linear groups, and related special functions. Izv. Akad. Nauk SSSR, Ser. Mat. 29, 1329–1356.Zbl. 185, 217.

    MathSciNet  MATH  Google Scholar 

  • Gel’fand, I.M., and Graev, M.I. (1965) Finite-dimensional irreducible representations of the unitary and full linear groups, and related special functions. English transl Am. Math. Soc. Gel’fand, I.M., and Graev, M.I. Transl., Ser. 2, 64, 116–146 (1967)

    Google Scholar 

  • Gel’fand, I.M., and Graev, M.I. (1986) A duality theorem for general hypergeometric functions. Dokl. Akad. Nauk SSSR 289, No. 1, 19–23.

    MathSciNet  Google Scholar 

  • Gel’fand, I.M., and Graev, M.I. (1986) A duality theorem for general hypergeometric functions. English transl.: Sov. Math., Dokl. 34, 9–13 (1987).Zbl. 619.33006

    MATH  Google Scholar 

  • Gel’fand, I.M., Graev, M.I., and Vilenkin, N.Ya. (1962) Integral Geometry and Related Problems of the Theory of Representations. Moscow: Fizmatgiz, 656 pp.Zbl. 115, 167. English transl.: New York: Academic Press (1966)

    Google Scholar 

  • Gel’fand, I.M., Graev, M.I., and Zelevinskij, A.V. (1987) Holonomic systems of equations and series of hypergeometric type. Dokl. Akad. Nauk SSSR 295, No. 1, 14–19.

    Google Scholar 

  • Gel’fand, I.M., Graev, M.I., and Zelevinskij, A.V. (1987) Holonomic systems of equations and series of hypergeometric type. English transl.: Sov. Math., Dokl. 36, 5–10 (1987).Zbl. 661.22005

    MATH  Google Scholar 

  • Gel’fand, I.M., and Zelevinskij, A.V. (1986) Algebraic and combinatorial aspects of the general theory of hypergeometric functions. Funk. Anal. Prilozh. 20, No. 3, 17–34.

    Google Scholar 

  • Gel’fand, I.M., and Zelevinskij, A.V. (1986) Algebraic and combinatorial aspects of the general theory of hypergeometric functions. English transl.: Funct. Anal. Appl. 20, 183–197 (1986).Zbl. 619.33004

    MATH  Google Scholar 

  • Gel’fand, I.M., Zelevinskij, A.V., and Kapranov, M.M. (1988) Equations of hypergeometric type and Newton polytopes. Dokl. Akad. Nauk SSSR 300, No. 3, 529–534.

    Google Scholar 

  • Gel’fand, I.M., Zelevinskij, A.V., and Kapranov, M.M. (1988) Equations of hypergeometric type and Newton polytopes. English transi.: Sov. Math., Dokl. 37, 678–682 (1988).Zbl. 667.33010

    MATH  Google Scholar 

  • Gel’fand, I.M., Minlos, R.A., and Shapiro, Z.Ya. (1958) Representations of the Rotation and Lorentz Groups and Their Applications. Moscow: Fizmatgiz. 336 pp. English transi.: New York: Macmillan (1963), XVIII, 366 pp.Zbl. 108, 220

    Google Scholar 

  • Gel’fand, I.M., and Naimark, M.A. (1950) Unitary representations of the classical groups. Tr. Mat. Inst. Steklova 36, 1288.

    MathSciNet  Google Scholar 

  • Gel’fand, I.M., and Naimark, M.A. (1950) Unitary representations of the classical groups. German transl.: Berlin: Akademie-Verlag (1957) XL, 333 pp. Zbl 77, 34

    Google Scholar 

  • Gel’fand, I.M., and Serganova, V.V. (1987) Combinatorial geometries and strats of torus on homogeneous manifolds. Usp. Mat. Nauk 42, No. 2, 107–134.

    MathSciNet  MATH  Google Scholar 

  • Gel’fand, I.M., and Serganova, V.V. (1987) Combinatorial geometries and strats of torus on homogeneous manifolds. English transi.: Russ. Math. Surv. 42, No. 2, 133–168 (1987).Zbl. 629.14035

    MathSciNet  MATH  Google Scholar 

  • Gel’fand, I.M., and Tsetlin, M.L. (1950) Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR 71, No. 5, 825–828.Zbl. 37, 153

    MATH  Google Scholar 

  • Gindikin, S.G. (1964) Analysis in homogeneous domains Usp. Mat. Nauk 19, No. 4, 3–92.Zbl. 144, 81

    MathSciNet  MATH  Google Scholar 

  • Godement, R. (1952) A theory of spherical functions. I. Trans. Am. Math. Soc. 73, 496–556.Zbl. 49, 201

    MathSciNet  MATH  Google Scholar 

  • Greiner, P.C. (1980) Spherical harmonics on the Heisenberg group. Can. Math. Bull. 23, 383–396.Zbl. 496.22012

    MathSciNet  MATH  Google Scholar 

  • Gross, K.I., and Holman, W.J. (1980) Matrix-valued special functions and representation theory of the conformal group. Trans. Am. Math. Soc. 258, 319–350.Zbl. 427.22008

    MathSciNet  MATH  Google Scholar 

  • Gross, K.I., Holman, W.J., and Kunze, R.A. (1977) The generalized gamma function, new Hardy spaces and representations of holomorphic type for the conformal group. Bull. Amer. Ma. Soc. 83, 412–415.Zbl. 349.22007

    MathSciNet  MATH  Google Scholar 

  • Gross, K.I., and Kunze, R.A. (1972) Fourier—Bessel transforms and holomorphic discrete series. Lect. Notes Math. 266, 79–122.Zbl. 242.22015

    MathSciNet  Google Scholar 

  • Gross, K.I., and Kunze, R.A. (1976) Bessel functions and representation theory. I. J. Funct. Anal. 22, 73–105.Zbl. 322.43014

    MathSciNet  MATH  Google Scholar 

  • Bessel functions and representation theory. II. J. Funct. Anal. 25, 1–49.Zbl. 361.22007

    Google Scholar 

  • Gruber, B., Doebner, H.D., and Feinsilver, P.J. (1982) Representations of the Heisenberg—Weyl algebra and group. Kinam. 4, 241278.Zbl. 502.22007

    Google Scholar 

  • Gustafson, R.A. (1987) A Whipple’s transformation for hypergeometric series in U(n) and multivariable hypergeometric orthogonal polynomials. SIAM J. Math. Anal 18, 495–530.Zbl. 607.33015

    MathSciNet  MATH  Google Scholar 

  • Gustafson, R.A., and Milne, S.C. (1983) Schur functions, Good’s identity, and hypergeometric series well—poised in SU(n). Adv. Math. 48, 177–188.Zbl. 516.33015

    MathSciNet  MATH  Google Scholar 

  • Harish—Chandra (1984) Collected Papers, vols. 1–4. Berlin Heidelberg New York: Springer. XLII, 2236 pp.Zbl. 546.01013

    Google Scholar 

  • Helgason, S. (1974) Eigenspaces of the Laplacian: integral representations and irreducibility. J. Funct. Anal. 17, 328–353.Zbl. 303.43021

    MathSciNet  MATH  Google Scholar 

  • Groups and Geometric Analysis. New York: Academic Press, 654 pp.Zbl. 543.58001

    Google Scholar 

  • Herz, C.S. (1955) Bessel functions of matrix argument. Ann. Math., II. Ser. 61, 474–523.Zbl. 66, 320

    MathSciNet  MATH  Google Scholar 

  • Holman, W.J., and Biedenharn, L.C. (1976) On hypergeometric series well-poised in SU(n). SIAM J. Math. Anal. 7, 529–541.Zbl. 329.33013

    MathSciNet  MATH  Google Scholar 

  • Hoogenboom, B. (1983) Intertwining Functions on Compact Lie Groups. Amsterdam: Mathematisch Centrum, 86 pp.Zbl. 553.43005

    Google Scholar 

  • Ikeda, M. (1967) On spherical functions for the unitary group. Mem. Fac. Engrg. Hiroshima 3, 17–75

    Google Scholar 

  • James, A.T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 40–75.Zbl. 55, 132

    MATH  Google Scholar 

  • James, A.T., and Constantine, A.G. (1974) Generalized Jacobi polynomials as spherical functions of the Grassmann manifold. Proc. Lond. Math. Soc., III. Ser. 29, 174–192.Zbl. 289.33031

    MathSciNet  MATH  Google Scholar 

  • Janson, S., and Peetre, J. (1987) A new generalization of Hankel operators. Math. Nadir. 132, 313–328.Zbl. 644.47026

    MathSciNet  MATH  Google Scholar 

  • Kac, V.G. (1974) Infinite-dimensional Lie algebras and Dedekind’s u-function. Funkts. Anal Prilozh. 8, No. 1, 77–78.

    Google Scholar 

  • Kac, V.G. (1974) Infinite-dimensional Lie algebras and Dedekind’s u-function. English transl.: Funct. Anal. Appl. 8, 68–70 (1974).Zbl. 299.17005

    MATH  Google Scholar 

  • Kac, V.G. (1983) Infinite-Dimensional Lie Algebras. Boston: Birkhäuser, 245 pp.Zbl. 537.17001

    MATH  Google Scholar 

  • Kalnins, E.G. (1973) Mixed basis matrix elements for the subgroup reduction of SO(2, 1). J. Math. Phys. 14, 654–657.Zbl. 255.22020

    MathSciNet  MATH  Google Scholar 

  • Karlin, S., and McGregor, J.L. (1961) The Hahn polynomials, formulas and an application. Scr. Math. 26, 33–46.Zbl. 104, 291

    MathSciNet  MATH  Google Scholar 

  • Kaufman, B. (1966) Special functions of mathematical physics from the viewpoint of the Lie algebra. J. Math. Phys. 7, 447–457.Zbl. 142, 35

    MathSciNet  MATH  Google Scholar 

  • Kil’dyushev, M.S. (1972) Hyperspherical functions of the “tree” type in the problem of n bodies. Yadernaya Fizika 15, 197–208.

    Google Scholar 

  • Kirillov, A.A. (1962) Unitary representations of nilpotent Lie groups. Usp. Mat. Nauk 17, No. 4, 57–110.

    MathSciNet  Google Scholar 

  • Kirillov, A.A. (1962) Unitary representations of nilpotent Lie groups. English transi Russ. Math. Surv. 17, No. 4, 53–104.Zbl. 106, 250

    MathSciNet  MATH  Google Scholar 

  • Kirillov, A.A. (1988) The theory of representations and non-commutative harmonic analysis. Itogi Nauki Tekh. Ser. Sovr. Probi. Mat. Fund. Napr. 22, 5–162.Zbl. 656.43005.

    MATH  Google Scholar 

  • Kirillov, A.A. (1988) English transi. in: Encyclopaedia of Mathematical Sciences, vol. 22. Berlin Heidelberg New York: Springer (1994)

    Google Scholar 

  • Klimyk, A.U. (1977) Matrix elements of irreducible representations of the group IU(n). Dokl. Akad. Nauk Ukr. SSR, Ser. A 12, 1103–1105.Zbl. 397.22005

    MathSciNet  Google Scholar 

  • Klimyk, A.U. (1979a) Matrix Elements and Clebsch-Gordan Coefficients of Group Representations. Kiev: Naukova Dumka (Russian)

    Google Scholar 

  • Klimyk, A.U. (1979b) Matrix elements and Casimir operators of the discrete series representations of U(p,q). Lett. Math. Phys. 3, 315–317.Zbl. 418.22018

    MathSciNet  MATH  Google Scholar 

  • Klimyk, A.U., and Gavrilik, A.M. (1979) Representation matrix elements and Clebsch—Gordan coefficients of semisimple Lie groups. J. Math. Phys. 20, 1624–1642.Zbl. 416.22017

    MathSciNet  MATH  Google Scholar 

  • Klink, W.H., and Ton-That, T. (1979) Orthogonal polynomial bases for holomorphically induced representations of the general linear groups. Ann. Inst. H. Poincaré., Sér. A 31, 99–113.Zbl. 439.22021

    MathSciNet  MATH  Google Scholar 

  • Klink, W.H., and Ton-That, T. (1982) Matrix elements and highest weight Wigner coefficients of GL(n, C). Ann. Inst. H. Poincaré. Sér. A 36, 225–237.Zbl. 488.22041

    MathSciNet  MATH  Google Scholar 

  • Klink, W.H., and Ton-That, T. (1986) Polynomial representations of the orthogonal groups. Acta Appl. Math. 6, 219–273.Zbl. 595.22018

    MathSciNet  MATH  Google Scholar 

  • Koornwinder, T.H. (1972) The addition formula for Jacobi polynomials Summary of results. Indag. Math. 34, 188–191.Zbl. 247.33017

    MathSciNet  Google Scholar 

  • Koornwinder, T.H. (1973) The addition formula for Jacobi polynomials and spherical harmonics. SIAM J. Appl. Math. 25, 236–246.Zbl. 276.33023

    MathSciNet  MATH  Google Scholar 

  • Koornwinder, T.H. (1975) A new proof of a Paley—Wiener type theorem for the Jacobi transform. Ark. Math. 13, 145–159.Zbl. 303.42022

    MathSciNet  MATH  Google Scholar 

  • Koornwinder, T.H. (1981) Clebsch—Gordan coefficients for SU(2) and Hahn polynomials. Nieuw. Arch. Wisk., III. Ser. 29, 140–155.Zbl. 483.20027

    MathSciNet  MATH  Google Scholar 

  • Koornwinder, T.H. (1982) Krawtchouk polynomials, a unification of two different group theoretical interpretations. SIAM J. Math. Anal 13, 1011–1023.Zbl. 505.33015

    MathSciNet  MATH  Google Scholar 

  • Koranyi, A. (1982) Kelvin transforms and harmonic polynomials on the Heisenberg group. J. Funct. Anal 49, 177–182.Zbl. 514.43006

    MathSciNet  MATH  Google Scholar 

  • Lebedev, N.N. (1963) Special Functions and Their Applications. Moscow: Fizmatgiz, 224 pp.Zbl. 52, 64.

    Google Scholar 

  • Lebedev, N.N. (1963) English transl.: Dover: Silvermann (1972)

    Google Scholar 

  • Lepowsky, J., and Wilson, R.L. (1982) A Lie theoretical interpretation and proof of the Rogers—Ramanujan identities. Adv. Math. 45, 21–72.Zbl. 488.17006

    MathSciNet  MATH  Google Scholar 

  • Lindblad, G. (1970) Eigenfunction expansions associated with unitary representations of SU(1, 1). Phys. Scr. 1, No. 5–6, 201–207

    MathSciNet  MATH  Google Scholar 

  • Louck, J.D., and Biedenharn, L.C. (1977) A generalization of the Gauss hypergeometric series. J. Math. Anal. Appl. 59, 423–431.Zbl. 354.33002

    MathSciNet  MATH  Google Scholar 

  • Maass, H. (1956) Spherical functions and quadratic forms. J. Indian Math. Soc., New Ser. 20, 117–162.Zbl. 72, 84

    Google Scholar 

  • Zur Theorie der Kugelfunktionen einer Matrixvariablen. Math. Ann. 137, 142–149.Zbl. 83, 60

    Google Scholar 

  • Macdonald, I.G. (1972) Affine root systems and Dedekind’s ti-function. Invent. Math. 15, 91–143.Zbl. 244.17005

    MathSciNet  MATH  Google Scholar 

  • McBride, E.B. (1971) Obtaining Generating Functions. Berlin Heidelberg New York: Springer. VIII, 100 pp.Zbl. 215, 434

    MATH  Google Scholar 

  • Mickelsson, J., and Niederle, J. (1973) On integrability of discrete representations of Lie algebra u(p, q). Ann. Inst. H. Poincaré. Ser. A 19, 171–177.Zbl. 285.22015

    MathSciNet  Google Scholar 

  • Miller, W., Jr. (1964) On Lie algebras and some special functions of mathematical physics. Mem. Am. Math. Soc. 50, 1–86.Zbl. 132, 296

    Google Scholar 

  • Miller, W., Jr. (1968a) Lie Theory and Special Functions. New York: Academic Press, 338 pp.Zbl. 174, 105

    MATH  Google Scholar 

  • Miller, W., Jr. (1968b) Special functions and the complex Euclidean group in 3-space. J. Math. Phys. 9, 1163–1175.Zbl. 159, 95

    MATH  Google Scholar 

  • Miller, W., Jr. (1972a) Lie theory and generalized hypergeometric functions. SIAM J. Math. Anal. 3, 31–44.Zbl. 235.33016

    MATH  Google Scholar 

  • Miller, W., Jr. (1972b) Clebsch-Gordan coefficients and special functions identities. I and II. J. Math. Phys. 13, 648–655

    MATH  Google Scholar 

  • Miller, W., Jr. (1972b) Clebsch-Gordan coefficients and special functions identities. I and II. J. Math. Phys. 13, 827–833.Zbl. 234.33017, Zbl. 235.33018

    MATH  Google Scholar 

  • Miller, W., Jr. (1972c) Lie theory and the Lauricella functions FD. J. Math. Phys. 13, 1393–1399.Zbl. 243.33009

    MATH  Google Scholar 

  • Miller, W., Jr. (1973) Lie theory and generalization of the hypergeometric functions. SIAM J. Appl. Math. 25, 226–235.Zbl. 269.33007

    MATH  Google Scholar 

  • Miller, W., Jr. (1978) Symmetry and Separation of Variables. Massachusetts: Addison-Wesley, 285 pp.Zbl. 368.35002

    Google Scholar 

  • Milne, S.C. (1980) Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn’s G function. Adv. Math. 36, 169–211.Zbl. 451.33010

    MathSciNet  MATH  Google Scholar 

  • Milne, S.C. (1985a) An elementary proof of the Macdonald identities for Al. Adv. Math. 57, 34–70.Zbl. 586.33011

    MathSciNet  MATH  Google Scholar 

  • Milne, S.C. (1985b) A new symmetry related to SU(n) for classical basic hypergeometric series. Adv. Math. 57, 71–90.Zbl. 586.33012

    MathSciNet  MATH  Google Scholar 

  • Milne, S.C. (1986) A U(n) generalization of Ramanujan’s 1W1 summation. J. Math. Anal. Appl. 118, 263–277.Zbl. 594.33008

    MathSciNet  MATH  Google Scholar 

  • Moody, R.V. (1975) Macdonald identities and Euclidean Lie algebras. Proc. Am. Math. Soc. 48, 43–52.Zbl. 315.17003

    MathSciNet  MATH  Google Scholar 

  • Nikiforov, A.F., Suslov, S.K., and Uvarov, V.B. (1985) Classical Orthogonal Polynomials of a Discrete Variable. Moscow: Nauka, 216 pp.Zbl. 576.33001. English transl.: Berlin Heidelberg New York: Springer (1991)

    Google Scholar 

  • Nikiforov, A.F., and Uvarov, V.B. (1984) Special Functions of Mathematical Physics. Moscow: Nauka, 344 pp.Zbl. 567.33001. English transl.: Basel: Birkhäuser (1988)

    Google Scholar 

  • Olevskij, M.N. (1949) On representation of an arbitrary function in the form of an integral with a kernel containing hypergeometric function. Dokl. Akad. Nauk SSSR 69, No. 1, 11–14.Zbl. 38, 216

    MATH  Google Scholar 

  • Paranuk, V.I. (1974a) On a certain class of functions of a matrix argument. Sb. Nauch. Tr. Mosk. Zaoch. Ped. Inst. 39, 139–164.

    Google Scholar 

  • Paranuk, V.I. (1974b) On properties of the function IMN(X). Sb. Nauch. Tr. Mosk. Zaoch. Ped. Inst. 39, 165–188

    Google Scholar 

  • Patera, J., and Winternitz, P.

    Google Scholar 

  • Paranuk, V.I. (1973) A new basis for the representations of the rotation group: Lamé and Heun polynomials. J. Math. Phys. 14, 1130–1139.Zbl. 285.22017

    Google Scholar 

  • Rozenblum, A.V. (1975)Matrix elements of class 1 irreducible representations of the group M(n) of motions of the n-dimensional Euclidean space. Dokl. Akad. Nauk BSSR 19, No. 4, 297–300.Zbl. 318.22018

    Google Scholar 

  • Rozenblum, A.V., and Rozenblum, L.V. (1985) Representations of the group SO(4) and Jacobi functions of matrix argument. Dokl. Akad. Nauk BSSR 29, 882–885

    Google Scholar 

  • (Russian) Zbl. 584.22006

    Google Scholar 

  • Rozenblum, A.V., and Rozenblum, L.V. (1986) Orthogonal polynomials of many variables connected with representations of the group of Euclidean motions. Differ. Uravn. 22, No. 11, 1961–1972

    Google Scholar 

  • Rozenblum, A.V., and Rozenblum, L.V. (1986) Orthogonal polynomials of many variables connected with representations of the group of Euclidean motions. English transi.: Differ. Equations 22, 1366–1375 (1986).Zbl. 629.33011

    Google Scholar 

  • Rozenblum, A.V., and Rozenblum, L.V. (1987) Representations of the group of rotations of Euclidean spaces and special functions of matrix indices. Dokl. Akad. Nauk SSSR 292, No. 2, 558–562.

    Google Scholar 

  • Rozenblum, A.V., and Rozenblum, L.V. (1987) Representations of the group of rotations of Euclidean spaces and special functions of matrix indices. English transl.: Sov. Math., Dokl. 35, 98–102 (1987).Zbl. 634.33020

    Google Scholar 

  • Schempp, W. (1979) Harmonic Analysis on the Heisenberg nilpotent Lie Group, with Applications to Signal Theory. Boston: Pitman, 199 pp.Zbl. 632.43001

    Google Scholar 

  • Schempp, W. (1984) Radar ambiguity functions, the Heisenberg group, and holomorphic theta series. Proc. Am. Math. Soc. 92, 103–110.Zbl. 541.43004

    MathSciNet  MATH  Google Scholar 

  • Shapiro, R.L. (1968) Special functions connected with representations of the group SU(n) of class 1 with respect to SU(n — 1). Izv. Vyssh. Uchebn. Zaved., Mat. 71, 97–107.Zbl. 179, 102

    Google Scholar 

  • Shlejnikova, M.A. (1974) Matrix elements of irreducible representations of the group SU(2, 1) and generalized hypergeometric function. Sb. Nauch. Tr. Mosk. Zaoch. Ped. Inst. 39, 219–236

    Google Scholar 

  • Slater, L.J. (1966) Generalized Hypergeometric Functions. Cambridge: Cambridge University Press, 273 pp.Zbl. 135, 281

    MATH  Google Scholar 

  • Smorodinskij, Ya.A., and Shelepin, L.A. (1972) Clebsch—Gordan coefficients viewed from different sides. Usp. Fiz. Nauk 106, No. 1, 3–45. English transl.: Sov. Phys. Usp. 15, 1–24 (1972)

    Google Scholar 

  • Smorodinskij, Ya.A., and Suslov, S.K. (1982) 6j-symbols and orthogonal polynomials. Yad. Fiz. 36, No. 4, 1066–1071

    Google Scholar 

  • Smorodinskij, Ya.A., and Suslov, S.K. (1982) 6j-symbols and orthogonal polynomials. English transi.: Soy. J. Nucl. Phys. 36, 623–625 (1982).Zbl. 586.22013

    Google Scholar 

  • Stanton, D. (1980) Some q-Krawtchouk polynomials on Chevalley groups. Amer. J. Math. 102, 625–662.Zbl. 448.33019

    MATH  Google Scholar 

  • Stanton, D. (1981) Three addition theorems for some q-Krawtchouk polynomials. Geom. Dedic. 10, 403–425.Zbl. 497.43006

    MATH  Google Scholar 

  • Stanton, D. (1984) Orthogonal polynomials and Chevalley groups, In: Special Functions: Group Theor. Aspects and Applications. Dordrecht: Reidel, 87–128.Zbl. 578.20041

    Google Scholar 

  • Stolov, E.L. (1974) Some inequalities for matrix elements of representations of SO(n). Sb. Nauch. Tr. Mosk. Zaoch. Ped. Inst. 39, 213–218 (Russian)

    Google Scholar 

  • Suslov, S.K. (1982) Bust matrix elements for the Lorentz group and Hahn orthogonal polynomials on a contour. Yad. Fiz. 36, No. 4, 1063–1065.

    MathSciNet  Google Scholar 

  • Suslov, S.K. (1982) Bust matrix elements for the Lorentz group and Hahn orthogonal polynomials on a contour. English transl.: Soy. J. Nucl. Phys. 36, 621–622 (1982).Zbl. 586.22012

    Google Scholar 

  • Suslov, S.K. (1983) 9j-symbols as orthogonal polynomials of two variables. Yad. Fiz. 38, No. 4, 1102–1104.

    Google Scholar 

  • Suslov, S.K. (1983) 9j-symbols as orthogonal polynomials of two variables. YadEnglish transi.: Sov. J. Nucl. Phys. 38, 662–662 (1983).Zbl. 592.33013

    MathSciNet  MATH  Google Scholar 

  • Szegö, G. (1959) Orthogonal Polynomials. New York: Am. Math. Soc.Zbl. 89, 275

    Google Scholar 

  • Talman, J (1968) Special Functions: a Group Theoretical Approach. New York: Benjamin. XII, 260 pp.Zbl. 197, 43

    Google Scholar 

  • Ton-That, T. (1976) Lie group representations and harmonic polynomials of a matrix variable. Trans. Am. Math. Soc. 216, 1–46.Zbl. 313.22010

    MATH  Google Scholar 

  • Truesdell, C. (1948) A Unified Theory of Special Functions. Princeton: Princeton Univ. Press. 182 pp.Zbl. 45, 343

    MATH  Google Scholar 

  • Varshalovich, D.A., Moskalev, A.I., and Khersonskij, V.K. (1975) Quantum Theory of Angular Momentum. Leningrad: Nauka, 440 pp. English transl.: Singapore: World Scientific Publ. (1989).Zbl. 725.00003

    Google Scholar 

  • Vasilev, V.A., Gel’fand, I.M., and Zelevinskij, A.V. (1987a) General hypergeometric functions on complex Grassmannians. Funkts. Anal. Prilozh. 21, No. 1, 23–38.Zbl. 614.33008.

    Google Scholar 

  • Vasilev, V.A., Gel’fand, I.M., and Zelevinskij, A.V. (1987a) English transi Funct. Anal. Appl. 21, 19–31 (1987)

    Google Scholar 

  • Vasilev, V.A., Gel’fand, I.M., and Zelevinskij, A.V. (1987b) Behavior of general hypergeometric functions in complex plane. Dokl. Akad. Nauk SSSR 290, No. 2, 277–281.

    Google Scholar 

  • Vasilev, V.A., Gel’fand, I.M., and Zelevinskij, A.V. (1987b) Behavior of general hypergeometric functions in complex plane. English transi.: Sov. Math., Dokl. 34, 268272 (1987).Zbl. 619.33005

    Google Scholar 

  • Vilenkin, N.Ya. (1957) On the theory of associated spherical functions on Lie groups. Mat. Sb., Nov. Ser. 42, 485–496.Zbl. 94, 247

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya. (1964a) The hypergeometric function and representations of the group of real matrices of the second order. Mat. Sb., Nov. Ser. 64, 497–520.

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya. (1964a) The hypergeometric function and representations of the group of real matrices of the second order. English transl.: Am. Math. Soc., Transi., II. Ser. 60, 159–184 (1967).Zbl. 198, 83

    Google Scholar 

  • Vilenkin, N.Ya. (1964b) Continual addition theorems for the hypergeometric function. Mat. Sb., Nov. Ser. 65, 28–46.Zbl. 135, 282

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya. (1965a) Polyspherical and orispherical functions. Mat. Sb., Nov. Ser. 68, 432–443.Zbl. 154, 368

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya. (1965b) Special Functions and the Theory of Group Representations. Moscow: Nauka, 588 pp.Zbl. 144, 380. English transi.: Providence: Am. Math. Soc. (1968)

    Google Scholar 

  • Vilenkin, N.Ya. (1967) Integral transforms of functions on hyperboloids. I. Mat. Sb., Nov. Ser. 74, 119–132.Zbl. 172, 355

    MathSciNet  Google Scholar 

  • Laguerre polynomials, Whittaker functions and representations of the group of bordered matrices. Mat. Sb., Nov. Ser. 75, 432–444 (Russian)

    Google Scholar 

  • Vilenkin, N.Ya. (1974) Matrix elements of irreducible unitary representations of the group U(n) and generalized Jacobi polynomials. Sb. Nauch. Tr. Mosk. Zaoch. Ped. Inst, 39, 77–90 (Russian)

    Google Scholar 

  • Vilenkin, N.Ya., Khudyakova, G.I., and Shapiro, R.L. (1977) Generalized Jacobi polynomials and their limit values. Funkts. Anal., 9, 29–39.Zbl. 467.43007

    MATH  Google Scholar 

  • Vilenkin, N.Ya., Klesova, L.M., and Pavlyuk, A.P. (1980) On the theory of representations of the pseudo-orthogonal group. In: Group Theoretical Methods in Physics. Moscow: Nauka, 40–46

    Google Scholar 

  • Vilenkin, N.Ya., and Klimyk, A.U. (1984) Wigner d-functions of the group U(3). Preprint No. 183. Inst. for Theoretical Physics, Kiev. 32 pp. (Russian)

    Google Scholar 

  • Vilenkin, N.Ya., and Klimyk, A.U. (1985a) Relations between associated spherical functions of symmetric spaces. Compact case. Preprint No. 136. Inst. for Theoretical Physics, Kiev. 32 pp. (Russian)

    Google Scholar 

  • Vilenkin, N.Ya., and Klimyk, A.U. (1985b) Relations between associated spherical functions of symmetric spaces. Non-compact case. Preprint No. 137. Inst. for Theoretical Physics, Kiev. 36 pp. (Russian)

    Google Scholar 

  • Vilenkin, N.Ya., and Klimyk, A.U. (1987) Associated spherical functions of pseudo-Riemannian symmetric spaces of rank 1. Dokl. Akad. Nauk Ukr. SSR, Ser. A 7, 9–12.Zbl. 629.33009

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya., and Klimyk, A.U. (1988a) Representations of the group SL(2, R) and relations for special functions of mathematical physics. Preprint No. 4, Inst. for Theoretical Physics, Kiev. 28 pp. (Russian)

    Google Scholar 

  • Vilenkin, N.Ya., and Klimyk, A.U. (1988b) Clebsch-Gordan coefficients, Racah coefficients, and polynomials of a discrete variable. Preprint No. 3, Inst. for Theoretical Physics, Kiev. 20 pp. (Russian)

    Google Scholar 

  • Vilenkin, N.Ya., and Nizhnikov, A.I. (1979) Integral relations for Mejer G-functions and representations of the n-dimensional Lorentz group. Izv. Vyssh. Ucheb. Zaved., Mat. 5, No. 1, 13–19.Zbl. 414.33003.

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya., and Nizhnikov, A.I. (1979) English transl.: Sov. Math. 23, No. 5, 11–17 (1979)

    MathSciNet  MATH  Google Scholar 

  • Vilenkin, N.Ya., and Paranuk, V.I. (1969) Group theoretical consideration of one class of generalized Bessel functions of a matrix argument. Sb. Nauch. Tr. Krasnodar. Ped. Inst. 118, 52–61

    MathSciNet  Google Scholar 

  • Vilenkin, N.Ya., and Smorodinskij, Ya.A. (1964) Invariant decompositions of relativistic amplitudes. Zhurn. Eksp. Teoret. Fiz. 46, 1793–1809

    Google Scholar 

  • Vilenkin, N.Ya., and Shapiro, R.L. (1967) Irreducible representations of the group U(n) of class 1 relative to SU(n - 1). Izv. Vyssh. Uchebn. Zaved., Mat. 62, 9–20.

    Google Scholar 

  • ilenkin, N.Ya., and Shapiro, R.L. (1967) Irreducible representations of the group U(n) of class 1 relative to SU(n - 1). English transl.: Am. Math. Soc. Transl. (2) 113, 187–200 (1979).Zbl. 408.22015

    Google Scholar 

  • Vilenkin, N.Ya., and Shlejnikova, M.A. (1970) Integral transforms for Whittaker functions and representations of three-dimensional Lorentz group. Mat. Sb., Nov. Ser. 81, 185–191.Zbl. 193, 360

    Google Scholar 

  • Vilenkin, N.Ya., and Shlejnikova, M.A. (1988) Integral transforms of functions and three-dimensional one-sheet hyperboloid. Funkts. Anal. 28, 47–54.Zbl. 667.44004

    Google Scholar 

  • Vretare, L. (1976) Elementary spherical functions on symmetric spaces. Math. Scand. 39, 343–358.Zbl. 387.43009

    MathSciNet  Google Scholar 

  • Vretare, L. (1984) Formulas for elementary spherical functions and generalized Jacobi polynomials. SIAM J. Math. Anal. 15, 805–833.Zbl. 549.43006

    MathSciNet  MATH  Google Scholar 

  • Warner, G. (1972) Harmonic Analysis on Semisimple Lie Groups. I. and II. Berlin Heidelberg New York: Springer.Zbl. 265.22020 and Zbl. 265.22021

    Google Scholar 

  • Weyl, H. (1931) The Theory of Groups and Quantum Mechanics. New York: Dover.Zbl. 1, 175

    MATH  Google Scholar 

  • Wigner, E.P. (1931) Gruppentheorie and ihre Anwenendung auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg.Zbl. 1, 175

    Google Scholar 

  • Wigner, E.P. (1955) The Application of Group Theory to the Special Functions of Mathematical Physics. Princeton lectures Wilson, J.A.

    Google Scholar 

  • Wigner, E.P. (1980) Some hypergeometric orthogonal polynomials SIAM J. Math. Anal 11, 690701.Zbl. 454.33007

    Google Scholar 

  • Yucys, A.P., Levinson, I.B., and Vanagas, V.V. (1960) Mathematical Means of the Theory of Angular Momentum. Vilnus: Mintis. English transi.: Jerusalem: Isr. Prog. Sci. Transi. (1962).Zbl. 111, 247

    Google Scholar 

  • Zhelobenko, D.P. (1970) Compact Lie Groups and Their Representations. Moscow: Nauka., 664 pp.

    Google Scholar 

  • Zhelobenko, D.P. (1970) English transi.: Providence: Amer. Math. Soc. (1973).Zbl. 228.22013

    Google Scholar 

  • Zhelobenko, D.P., and Shtern, A.I. (1983) Representations of Lie Groups. Moscow: Nauka.Zbl. 521.22006

    MATH  Google Scholar 

Download references

Authors
  1. A. U. Klimyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Ya. Vilenkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, 19104-6395, Philadelphia, PA, USA

    A. A. Kirillov

  2. Moscow University, Mehmat, 117234, Moscow, Russia

    A. A. Kirillov

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Klimyk, A.U., Vilenkin, N.Y. (1995). Representations of Lie Groups and Special Functions. In: Kirillov, A.A. (eds) Representation Theory and Noncommutative Harmonic Analysis II. Encyclopaedia of Mathematical Sciences, vol 59. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09756-4_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-09756-4_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08126-2

  • Online ISBN: 978-3-662-09756-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation