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Volumn 13, Issue 1-3, 2007, Pages 131-166

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Author keywords

Biorthogonal rational function; Eight vertex model; Elliptic 6j symbol; Elliptic hypergeometric series; Sklyanin algebra

Indexed keywords


EID: 33845728851     PISSN: 13824090     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11139-006-0245-1     Document Type: Conference Paper
Times cited : (56)

References (64)
  • 1
    • 84966223268 scopus 로고
    • A q-beta integral on the unit circle and some biorthogonal rational functions
    • Al-Salam, W.A., Ismail, M.E.H.: A q-beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121, 553-561 (1994)
    • (1994) Proc. Amer. Math. Soc. , vol.121 , pp. 553-561
    • Al-Salam, W.A.1    Ismail, M.E.H.2
  • 2
    • 0039050294 scopus 로고
    • q-Analogs of some biorthogonal functions
    • Al-Salam, W.A., Verma, A.: q-Analogs of some biorthogonal functions. Canad. Math. Bull. 26, 225-227 (1983)
    • (1983) Canad. Math. Bull. , vol.26 , pp. 225-227
    • Al-Salam, W.A.1    Verma, A.2
  • 3
    • 0021422438 scopus 로고
    • Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities
    • Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Statist. Phys. 35, 193-266 (1984)
    • (1984) J. Statist. Phys. , vol.35 , pp. 193-266
    • Andrews, G.E.1    Baxter, R.J.2    Forrester, P.J.3
  • 4
    • 0001066491 scopus 로고
    • A set of orthogonal polynomials that generalize the Racah coefficients or 6- j symbols
    • Askey, R., Wilson, J.A.: A set of orthogonal polynomials that generalize the Racah coefficients or 6- j symbols. SIAM J. Math. Anal. 10, 1008-1016 (1979)
    • (1979) SIAM J. Math. Anal. , vol.10 , pp. 1008-1016
    • Askey, R.1    Wilson, J.A.2
  • 5
    • 0001308382 scopus 로고
    • Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
    • Askey, R., Wilson, J.A.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 54(319) (1985)
    • (1985) Mem. Amer. Math. Soc. , vol.54 , Issue.319
    • Askey, R.1    Wilson, J.A.2
  • 6
    • 0001181936 scopus 로고
    • Universal exchange algebra for Bloch waves and Liouville theory
    • Babelon, O.: Universal exchange algebra for Bloch waves and Liouville theory. Comm. Math. Phys. 139, 619-643 (1991)
    • (1991) Comm. Math. Phys. , vol.139 , pp. 619-643
    • Babelon, O.1
  • 7
    • 49549172862 scopus 로고
    • Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II. Equivalence to a generalized ice-type model
    • Baxter, R.J.: Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain II. Equivalence to a generalized ice-type model. Ann. Phys. 76, 25-47 (1973)
    • (1973) Ann. Phys. , vol.76 , pp. 25-47
    • Baxter, R.J.1
  • 9
    • 0003192389 scopus 로고    scopus 로고
    • Exactly solvable SOS models II. Proof of the star-triangle relation and combinatorial identities
    • Jimbo, M., et al. (eds.)
    • Date, E.M., Jimbo, A., Kuniba, T., Miwa, M., Okado.: Exactly solvable SOS models II. Proof of the star-triangle relation and combinatorial identities. In: Jimbo, M., et al. (eds.) Conformal Field Theory and Solvable Lattice Models, pp. 17-122,
    • Conformal Field Theory and Solvable Lattice Models , pp. 17-122
    • Date, E.M.1    Jimbo, A.2    Kuniba, T.3    Miwa, M.4    Okado5
  • 10
    • 0007029026 scopus 로고
    • Academic Press, Boston, MA
    • Adv. Stud. Pure Math. 16, Academic Press, Boston, MA (1988)
    • (1988) Adv. Stud. Pure Math. , pp. 16
  • 11
    • 33744723844 scopus 로고
    • Fusion of the eight vertex SOS model
    • Erratum and addendum. Lett. Math. Phys. 14, 97 (1987)
    • Date, E., Jimbo, M., Miwa, T., Okado, M.: Fusion of the eight vertex SOS model. Lett. Math. Phys. 12 (1986), 209-215; Erratum and addendum. Lett. Math. Phys. 14, 97 (1987)
    • (1986) Lett. Math. Phys. , vol.12 , pp. 209-215
    • Date, E.1    Jimbo, M.2    Miwa, T.3    Okado, M.4
  • 12
    • 0003989383 scopus 로고    scopus 로고
    • Lectures on representation theory and knizhnikzamolodchikov equations
    • Providence, RI
    • Etingof, P.I., Frenkel, I. B., Kirillov, A. A.: Lectures on representation theory and knizhnikzamolodchikov equations. Amer. Math. Soc., Providence, RI (1998)
    • (1998) Amer. Math. Soc.
    • Etingof, P.I.1    Frenkel, I.B.2    Kirillov, A.A.3
  • 14
    • 33845788949 scopus 로고
    • Trigonometric solutions of the Yang-Baxter equation, nets, and hypergeometric functions
    • Gindikin, S., et al. (eds.) Birkhäuser, Boston
    • Frenkel, I.B., Turaev, V.G.: Trigonometric solutions of the Yang-Baxter equation, nets, and hypergeometric functions. In: Gindikin, S., et al. (eds.) Functional Analysis on the Eve of the 21st Century, vol. 1, pp. 65-118, Birkhäuser, Boston (1995)
    • (1995) Functional Analysis on the Eve of the 21st Century , vol.1 , pp. 65-118
    • Frenkel, I.B.1    Turaev, V.G.2
  • 15
    • 0001432807 scopus 로고    scopus 로고
    • Elliptic solutions of theYang-Baxter equation and modular hypergeometric functions
    • Arnold, V.I., et al. (eds.) Birkhäuser, Boston
    • Frenkel, I.B., Turaev, V. G.: Elliptic solutions of theYang-Baxter equation and modular hypergeometric functions. In: Arnold, V.I., et al. (eds.) The Arnold-Gelfand Mathematical Seminars, pp. 171-204. Birkhäuser, Boston (1997)
    • (1997) The Arnold-Gelfand Mathematical Seminars , pp. 171-204
    • Frenkel, I.B.1    Turaev, V.G.2
  • 17
    • 21944434494 scopus 로고    scopus 로고
    • Contiguous relations, continued fractions and orthogonality
    • Gupta, D.P., Masson, D.R.: Contiguous relations, continued fractions and orthogonality. Trans. Amer. Math. Soc. 350, 769-808 (1998)
    • (1998) Trans. Amer. Math. Soc. , vol.350 , pp. 769-808
    • Gupta, D.P.1    Masson, D.R.2
  • 18
    • 0003081035 scopus 로고
    • q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals
    • Ismail, M.E.H., Masson, D.R.: q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals. Trans. Amer. Math. Soc. 346, 63-116 (1994)
    • (1994) Trans. Amer. Math. Soc. , vol.346 , pp. 63-116
    • Ismail, M.E.H.1    Masson, D.R.2
  • 19
    • 0141820548 scopus 로고
    • Generalized orthogonality and continued fractions
    • Ismail, M.E.H., Masson, D.R.: Generalized orthogonality and continued fractions. J. Approx. Theory 83, 1-40 (1995)
    • (1995) J. Approx. Theory , vol.83 , pp. 1-40
    • Ismail, M.E.H.1    Masson, D.R.2
  • 21
    • 0001311498 scopus 로고
    • Summation of q-hypergeometric series
    • Jackson, F.H.: Summation of q-hypergeometric series. Messenger of Math. 50 (1921), 101-112.
    • (1921) Messenger of Math. , vol.50 , pp. 101-112
    • Jackson, F.H.1
  • 22
    • 0002444027 scopus 로고
    • On coherent states for the simplest quantum groups
    • B. Jurčo,. On coherent states for the simplest quantum groups. Lett. Math. Phys. 21, 51-58 (1991)
    • (1991) Lett. Math. Phys. , vol.21 , pp. 51-58
    • Jurčo, B.1
  • 23
    • 0001185834 scopus 로고
    • Representations of the algebraUq (sl(2)), q-orthogonal polynomials and invariants of links
    • Kac, V.G. (ed.) World Sci. Publ. Co., Teaneck, NJ
    • Kirillov, A.N., Reshetikhin, N.Yu.: Representations of the algebraUq (sl(2)), q-orthogonal polynomials and invariants of links. In: Kac, V.G. (ed.) Infinite-dimensional Lie Algebras and Groups, pp. 285-339. World Sci. Publ. Co., Teaneck, NJ (1989)
    • (1989) Infinite-dimensional Lie Algebras and Groups , pp. 285-339
    • Kirillov, A.N.1    Reshetikhin, N.Y.2
  • 25
    • 0000821927 scopus 로고    scopus 로고
    • Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications
    • Koelink, H.T.: Askey-Wilson polynomials and the quantum SU(2) group: survey and applications. Acta Appl. Math. 44, 295-352 (1996)
    • (1996) Acta Appl. Math. , vol.44 , pp. 295-352
    • Koelink, H.T.1
  • 26
    • 1842538819 scopus 로고    scopus 로고
    • Elliptic U(2) quantum group and elliptic hypergeometric series
    • Koelink, E., van Norden, Y., Rosengren, H.: Elliptic U(2) quantum group and elliptic hypergeometric series. Comm. Math. Phys. 245, 519-537 (2004)
    • (2004) Comm. Math. Phys. , vol.245 , pp. 519-537
    • Koelink, E.1    Van Norden, Y.2    Rosengren, H.3
  • 27
    • 0032383984 scopus 로고    scopus 로고
    • Convolutions for orthogonal polynomials from Lie and quantum algebra representations
    • 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)
    • (1998) SIAM J. Math. Anal. , vol.29 , pp. 794-822
    • Koelink, H.T.1    Van Der Jeugt, J.2
  • 28
    • 0005318334 scopus 로고
    • Krawtchouk polynomials, a unification of two different group theoretic interpretations
    • Koornwinder, T.H.: Krawtchouk polynomials, a unification of two different group theoretic interpretations. SIAM J. Math. Anal. 13, 1011-1023 (1982)
    • (1982) SIAM J. Math. Anal. , vol.13 , pp. 1011-1023
    • Koornwinder, T.H.1
  • 29
    • 0009464049 scopus 로고
    • Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group
    • Koornwinder, T.H.: Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group. SIAM J. Math. Anal. 24, 795-813 (1993)
    • (1993) SIAM J. Math. Anal. , vol.24 , pp. 795-813
    • Koornwinder, T.H.1
  • 30
    • 33845753222 scopus 로고    scopus 로고
    • A second addition formula for continuous q-ultraspherical polynomials
    • Koelink, E., Ismail, M.E.H. (eds.) Kluwer Acad. Publ.
    • Koornwinder, T.H.: A second addition formula for continuous q-ultraspherical polynomials. In:Koelink, E., Ismail, M.E.H. (eds.) Theory and Applications of Special Functions. A Volume Dedicated to Mizan Rahman, pp. 339-260. Kluwer Acad. Publ. (2005)
    • (2005) Theory and Applications of Special Functions. A Volume Dedicated to Mizan Rahman , pp. 339-1260
    • Koornwinder, T.H.1
  • 32
    • 0001282924 scopus 로고
    • Orthogonal polynomials, duality and association schemes
    • Leonard, D.A.: Orthogonal polynomials, duality and association schemes. SIAM J. Math. Anal. 13, 656-663 (1982)
    • (1982) SIAM J. Math. Anal. , vol.13 , pp. 656-663
    • Leonard, D.A.1
  • 33
    • 0009122874 scopus 로고
    • Unformung gewisser Reihen, deren Glieder Produkte hypergeometrische Funktionen sind
    • Meixner, J.: Unformung gewisser Reihen, deren Glieder Produkte hypergeometrische Funktionen sind. Deutsche Math. 6, 341-349 (1941)
    • (1941) Deutsche Math. , vol.6 , pp. 341-349
    • Meixner, J.1
  • 34
    • 0022187414 scopus 로고
    • Orthogonal polynomials and some q-beta integrals of Ramanujan
    • Pastro, P.I.: Orthogonal polynomials and some q-beta integrals of Ramanujan. J. Math. Anal. Appl. 112, 517-540 (1985)
    • (1985) J. Math. Anal. Appl. , vol.112 , pp. 517-540
    • Pastro, P.I.1
  • 35
    • 30244453087 scopus 로고
    • Theory of complex spectra II
    • G. Racah,. Theory of complex spectra II. Phys. Rev. 62, 438-462 (1942)
    • (1942) Phys. Rev. , vol.62 , pp. 438-462
    • Racah, G.1
  • 36
    • 0040379124 scopus 로고
    • Families of biorthogonal rational functions in a discrete variable
    • Rahman, M.: Families of biorthogonal rational functions in a discrete variable. SIAM J. Math. Anal. 12, 355-367 (1981)
    • (1981) SIAM J. Math. Anal. , vol.12 , pp. 355-367
    • Rahman, M.1
  • 37
    • 0000946162 scopus 로고
    • An integral representation of a 10.9 and continuous bi-orthogonal 10.9 rational functions
    • Rahman, M.: An integral representation of a 10.9 and continuous bi-orthogonal 10.9 rational functions. Canad. J. Math. 38, 605-618 (1986)
    • (1986) Canad. J. Math. , vol.38 , pp. 605-618
    • Rahman, M.1
  • 38
    • 33845721541 scopus 로고
    • Some extensions of Askey-Wilson's q-beta integral and the corresponding orthogonal systems
    • Rahman, M.: Some extensions of Askey-Wilson's q-beta integral and the corresponding orthogonal systems. Canad. Math. Bull. 31, 467-476 (1988)
    • (1988) Canad. Math. Bull. , vol.31 , pp. 467-476
    • Rahman, M.1
  • 39
    • 0001262738 scopus 로고
    • Biorthogonality of a system of rational functions with respect to a positive measure on [-1, 1]
    • Rahman, M.: Biorthogonality of a system of rational functions with respect to a positive measure on [-1, 1]. SIAM J. Math. Anal. 22, 1430-1441 (1991)
    • (1991) SIAM J. Math. Anal. , vol.22 , pp. 1430-1441
    • Rahman, M.1
  • 41
    • 0004034109 scopus 로고
    • Springer-Verlag, Berlin
    • Lecture Notes in Math. 1550, Springer-Verlag, Berlin (1993)
    • (1993) Lecture Notes in Math , pp. 1550
  • 44
    • 0011332853 scopus 로고    scopus 로고
    • A new quantum algebraic interpretation of the Askey-Wilson polynomials
    • H. Rosengren, A new quantum algebraic interpretation of the Askey-Wilson polynomials. Contemp. Math. 254, 371-394 (2000)
    • (2000) Contemp. Math. , vol.254 , pp. 371-394
    • Rosengren, H.1
  • 45
    • 0346499268 scopus 로고    scopus 로고
    • Elliptic hypergeometric series on root systems
    • Rosengren, H.: Elliptic hypergeometric series on root systems. Adv. Math. 181, 417-447 (2004)
    • (2004) Adv. Math. , vol.181 , pp. 417-447
    • Rosengren, H.1
  • 46
    • 34250143502 scopus 로고
    • Some algebraic structures connected with the Yang-Baxter equation
    • Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Functional Anal. Appl. 16, 263-270 (1982)
    • (1982) Functional Anal. Appl. , vol.16 , pp. 263-270
    • Sklyanin, E.K.1
  • 47
    • 0040694601 scopus 로고
    • Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras
    • Sklyanin, E.K.: Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras. Functional Anal. Appl. 17, 273-284 (1983)
    • (1983) Functional Anal. Appl. , vol.17 , pp. 273-284
    • Sklyanin, E.K.1
  • 48
    • 0036039842 scopus 로고    scopus 로고
    • An elliptic incarnation of the Bailey chain
    • Spiridonov, V.P.: An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 37, 1945-1977 (2002)
    • (2002) Int. Math. Res. Not. , vol.37 , pp. 1945-1977
    • Spiridonov, V.P.1
  • 49
  • 50
    • 3843146673 scopus 로고    scopus 로고
    • Theta hypergeometric integrals
    • Spiridonov, V.P.: Theta hypergeometric integrals. Algebra i Analiz 15, 161-215 (2003)
    • (2003) Algebra i Analiz , vol.15 , pp. 161-215
    • Spiridonov, V.P.1
  • 51
    • 0034349855 scopus 로고    scopus 로고
    • Spectral transformation chains and some new biorthogonal rational functions
    • Spiridonov, V.P., Zhedanov, A.S.: Spectral transformation chains and some new biorthogonal rational functions. Comm. Math. Phys. 210, 49-83 (2000)
    • (2000) Comm. Math. Phys. , vol.210 , pp. 49-83
    • Spiridonov, V.P.1    Zhedanov, A.S.2
  • 52
    • 0242575584 scopus 로고    scopus 로고
    • Generalized eigenvalue problems and a new family of rational functions biorthogonal on elliptic grids
    • Bustoz, J., et al. (eds.) Kluwer Acad. Publ., Dordrecht
    • Spiridonov, V.P., Zhedanov, A.S.: Generalized eigenvalue problems and a new family of rational functions biorthogonal on elliptic grids. In: Bustoz, J., et al. (eds.) Special Functions 2000: Current Perspective and Future Directions, pp. 365-388. Kluwer Acad. Publ., Dordrecht (2001)
    • (2001) Special Functions 2000: Current Perspective and Future Directions , pp. 365-388
    • Spiridonov, V.P.1    Zhedanov, A.S.2
  • 53
    • 1042268352 scopus 로고    scopus 로고
    • Vertex-IRF transformations, dynamical quantum groups and harmonic analysis
    • Stokman, J.V.: Vertex-IRF transformations, dynamical quantum groups and harmonic analysis. Indag. Math. 14, 545-570 (2003)
    • (2003) Indag. Math. , vol.14 , pp. 545-570
    • Stokman, J.V.1
  • 54
    • 0000047282 scopus 로고
    • Bethe ansatz for higher spin eight-vertex models
    • Takebe, T.: Bethe ansatz for higher spin eight-vertex models. J. Phys. A 28, 6675-6706 (1995)
    • (1995) J. Phys. A , vol.28 , pp. 6675-6706
    • Takebe, T.1
  • 56
    • 0036624901 scopus 로고    scopus 로고
    • Leonard pairs from 24 points of view
    • Terwilliger, P.: Leonard pairs from 24 points of view. Rocky Mountain J. Math. 32, 827-888 (2002)
    • (2002) Rocky Mountain J. Math. , vol.32 , pp. 827-888
    • Terwilliger, P.1
  • 60
    • 0003243132 scopus 로고
    • On the matrices which reduce the Kronecker products of representations of S. R. groups
    • manuscript (1940), published Biedenharn, L.C., Van Dam, H. (eds.) Academic Press, New York
    • Wigner, E.P.: On the matrices which reduce the Kronecker products of representations of S. R. groups, manuscript (1940), published. In: Biedenharn, L.C., Van Dam, H. (eds.) Quantum Theory of Angular Momentum, pp. 87-133. Academic Press, New York (1965)
    • (1965) Quantum Theory of Angular Momentum , pp. 87-133
    • Wigner, E.P.1
  • 62
    • 0000102490 scopus 로고
    • Orthogonal functions from Gram determinants
    • Wilson, J.A.: Orthogonal functions from Gram determinants. SIAM J. Math. Anal. 22, 1147-1155 (1991)
    • (1991) SIAM J. Math. Anal. , vol.22 , pp. 1147-1155
    • Wilson, J.A.1
  • 64
    • 1542755349 scopus 로고
    • "Hidden symmetry" of Askey-Wilson polynomials
    • Zhedanov, A.S.: "Hidden symmetry" of Askey-Wilson polynomials. Theoret. Math. Phys. 89, 1146-1157 (1991)
    • (1991) Theoret. Math. Phys. , vol.89 , pp. 1146-1157
    • Zhedanov, A.S.1


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