Journal of Mathematical Sciences

, Volume 213, Issue 5, pp 786–794 | Cite as

Generating Functions of Chebyshev Polynomials in Three Variables

  • M. A. SokolovEmail author

In this paper, we obtain generating functions of three-variable Chebyshev polynomials (of the first as well as of the second type) associated with the root system of the A3 Lie algebra. Bibliography: 21 titles.


Orthogonal Polynomial Weyl Group Simple Root Chebyshev Polynomial Fundamental Weight 
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  1. 1.
    T. N. Koornwinder, “Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I–IV,” Indagationes Mathematicae Proc., 77, 48–66, 357–81 (1974).CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    G. J. Heckman, “Root systems and hypergeometric functions. II,” Comp. Math., 64, 353–73 (1987).MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. E. Hoffman and W. D. Withers, “Generalized Chebyshev polynomials associated with affine Weyl groups,” Trans. Amer. Math. Soc., 308, 91–104 (1988).CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    R. J. Beerends, “Chebyshev polynomials in several variables and the radial part Laplace–Beltrami operator,” Trans. Amer. Math. Soc., 328, 770–814 (1991).CrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Klimyk and J. Patera, “Orbit functions,” SIGMA, 2, 006 (2006).MathSciNetGoogle Scholar
  6. 6.
    V. D. Lyakhovsky and P. V. Uvarov, “Multivariate Chebyshev polynomials,” J. Phys. A: Math. Theor., 46, 125201 (2013).CrossRefMathSciNetGoogle Scholar
  7. 7.
    B. N. Ryland and H. Z. Munthe-Kaas, “On multivariate Chebyshev polynomials and spectral approximations on triangles. Spectral and high order methods for partial differential equations,” Lect. Notes Comput. Sci. Eng., 76, Berlin, Springer, 19–41 (2011).Google Scholar
  8. 8.
    B. Shapiro and M. Shapiro, “On eigenvalues of rectangular matrices,” Trudy MIAN, 267, 258–265 (2009).Google Scholar
  9. 9.
    P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Multiplicity function for tensor powers of modules of the A n algebra,” Teor. Mat. Fiz., 171, 283–293 (2012).CrossRefMathSciNetGoogle Scholar
  10. 10.
    P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Tensor power decomposition. B n -case,” J. of Physics: Conference Series, 343, 012095 (2012).Google Scholar
  11. 11.
    P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Tensor powers for non-simply laced Lie algebras. B 2 -case,” J. of Physics: Conference Series, 346, 012012 (2012).Google Scholar
  12. 12.
    V. D. Lyakhovsky, “Multivariate Chebyshev polynomials in terms of singular elements,” Teor. Mat. Fiz., 175, 797–805 (2013).CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    V. V. Borzov and E. V. Damaskinsky, “Chebyshev–Koornwinder oscillator,” Teor. Mat. Fiz., 175, 765–772 (2013).CrossRefMathSciNetGoogle Scholar
  14. 14.
    V. V. Borzov and E. V. Damaskinsky, “The algebra of two-dimensional generalized Chebyshev-Koornwinder oscillator,” J. of Math. Physics, 55, 103505 (2014).CrossRefMathSciNetGoogle Scholar
  15. 15.
    G. Von Gehlen and S. Roan, “The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials,” in: S. Pakuliak and G. Von Gehlen (eds), Integrable sturctures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000), NATO Sci. Ser. II, Math. Phys. Chem., 35, Kluwer (2001), pp. 155–172.Google Scholar
  16. 16.
    G. Von Gehlen, “2002 Onsager’s algebra and partially orthogonal polynomials,” Int. J. Mod. Phys., B 16, 2129.CrossRefGoogle Scholar
  17. 17.
    P. K. Suetin, Classical Orthogonal Polynomials [in Russian], Nauka, Moscow (1979).Google Scholar
  18. 18.
    N. Bourbaki, 1975 Elements de Mathematique. Groupes et Algebres de Lie Hermann, Paris (1975).Google Scholar
  19. 19.
    B. Ken Dunn and R. Lidl, “Generalizations of the classical Chebyshev polynomials to polynomials in two variables,” Czech. Math. J., 32, 516–528 (1982).Google Scholar
  20. 20.
    E. V. Damaskinsky, P. P. Kulish, and M. A. Sokolov, “On calculation of generating functions of multivariate Chebyshev polynomials,” POMI preprint 13/2014.Google Scholar
  21. 21.
    JiaChang Sun, “A new class of three-variable orthogonal polynomials and their recurrences relations,” Science in China, Series A: Mathematics, 51, 1071–1092 (2008).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St.Petersburg State Polytechnical UniversitySt.PetersburgRussia

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