Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator

Author: R. J. Beerends
Journal: Trans. Amer. Math. Soc. 328 (1991), 779-814
MSC: Primary 33C45; Secondary 22E30, 33C80, 43A85
MathSciNet review: 1019520
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Chebyshev polynomials of the first and the second kind in $ n$ variables $ {z_1},{z_2}, \ldots,{z_n}$ are introduced. The variables $ {z_1},{z_2}, \ldots,{z_n}$ are the characters of the representations of $ SL(n + 1,{\mathbf{C}})$ corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates $ {z_1},{z_2}, \ldots,{z_n}$ and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.

References [Enhancements On Off] (What's this?)

  • [1] H. Bacry, Generalized Chebyshev polynomials and characters of $ GL(N,{\mathbf{C}})$ and $ SL(N,{\mathbf{C}})$, Group Theoretical Methods in Physics, Lecture Notes in Phys., vol. 201, Springer-Verlag, Berlin, 1984.
  • [2] -, An application of Laguerre's emanent to generalized Chebyshev polynomials, Polynômes Orthogonaux et Applications, Lecture Notes in Math., vol. 1171, Springer-Verlag, Berlin, 1985.
  • [3] Henri Bacry, Zeros of polynomials and generalized Chebyshev polynomials, Group-theoretic methods in physics, Vol. 2 (Russian) (Jūrmala, 1985) “Nauka”, Moscow, 1986, pp. 239–249. MR 946912
  • [4] R. J. Beerends, The Abel transform and shift operators, Compositio Math. 66 (1988), no. 2, 145–197. MR 945549
  • [5] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [6] -, Éléments de mathématique. Groupes et algèbres de Lie, Chapitres 7 et 8, Hermann, Paris, 1975.
  • [7] Amédée Debiard, Polynômes de Tchébychev et de Jacobi dans un espace euclidien de dimension 𝑝, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 13, 529–532 (French, with English summary). MR 703221
  • [8] Amédée Debiard, Système différentiel hypergéométrique de type 𝐵𝐶_{𝑝}, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 13, 363–366 (French, with English summary). MR 889739,
  • [9] -, Système différentiel hypergéométrique et parties radiales des opérateurs invariants des espaces symétriques de type $ B{C_p}$, Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math., vol. 1296, Springer-Verlag, Berlin, 1987.
  • [10] Amédée Debiard and Bernard Gaveau, Analysis on root systems, Canad. J. Math. 39 (1987), no. 6, 1281–1404. MR 918384,
  • [11] K. B. Dunn and R. Lidl, Multidimensional generalizations of the Chebyshev polynomials. I, II, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), no. 4, 154–159, 160–165. MR 575995
  • [12] Ken B. Dunn and Rudolf Lidl, Generalizations of the classical Chebyshev polynomials to polynomials in two variables, Czechoslovak Math. J. 32(107) (1982), no. 4, 516–528. MR 682129
  • [13] Richard Eier, Rudolf Lidl, and Ken B. Dunn, Differential equations for generalized Chebyshev polynomials, Rend. Mat. (7) 1 (1981), no. 4, 633–646 (English, with Italian summary). MR 647460
  • [14] Richard Eier and Rudolf Lidl, A class of orthogonal polynomials in 𝑘 variables, Math. Ann. 260 (1982), no. 1, 93–99. MR 664368,
  • [15] G. J. Heckman, Root systems and hypergeometric functions. II, Compositio Math. 64 (1987), no. 3, 353–373. MR 918417
  • [16] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
  • [17] Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
  • [18] Tom H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 48–58. MR 0340673
  • [19] Rudolf Lidl, Tschebyscheffpolynome in mehreren Variablen, J. Reine Angew. Math. 273 (1975), 178–198 (German). MR 0364200,
  • [20] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR 553598
  • [21] I. G. Macdonald, Commuting differential operators and zonal spherical functions, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 189–200. MR 911140,
  • [22] E. M. Opdam, Generalized hypergeometric functions associated with root systems, Thesis, Leiden, 1988.
  • [23] P. E. Ricci, I polinomi di Tchebycheff in più variabili, Rend. Mat. 11 (1978), 295-327.
  • [24] Jirō Sekiguchi, Zonal spherical functions on some symmetric spaces, Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976), 1976/77 supplement, pp. 455–459. MR 0461040,
  • [25] J. J. Sylvester, On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure, The Collected Mathematical Papers, vol. I, Chelsea, New York, 1973.
  • [26] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Prentice-Hall Series in Modern Analysis. MR 0376938
  • [27] Lars Vretare, Formulas for elementary spherical functions and generalized Jacobi polynomials, SIAM J. Math. Anal. 15 (1984), no. 4, 805–833. MR 747438,
  • [28] H. Weber, Lehrbuch der Algebra, Zweite auflage, Vieweg, Braunschweig, 1898.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 33C45, 22E30, 33C80, 43A85

Retrieve articles in all journals with MSC: 33C45, 22E30, 33C80, 43A85

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society