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1: 18.36 Miscellaneous Polynomials
§18.36(iv) Orthogonal Matrix Polynomials
These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. …
2: Bibliography D
  • P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
  • A. J. Durán and F. A. Grünbaum (2005) A survey on orthogonal matrix polynomials satisfying second order differential equations. J. Comput. Appl. Math. 178 (1-2), pp. 169–190.
  • 3: 18.38 Mathematical Applications
    Random Matrix Theory
    4: Bibliography B
  • P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.
  • 5: 35.4 Partitions and Zonal Polynomials
    6: 18.2 General Orthogonal Polynomials
    §18.2 General Orthogonal Polynomials
    7: 35.1 Special Notation
    a , b complex variables.
    𝟎 zero matrix.
    𝐇 orthogonal matrix.
    The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( 𝐓 ) and (of the second kind) B ν ( 𝐓 ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; 𝐓 ) or F 1 1 ( a b ; 𝐓 ) and (of the second kind) Ψ ( a ; b ; 𝐓 ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; 𝐓 ) or F 1 2 ( a 1 , a 2 b ; 𝐓 ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; 𝐓 ) or F q p ( a 1 , , a p b 1 , , b q ; 𝐓 ) . … Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( 𝐓 ) = A ν ( 𝐓 ) / A ν ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | 𝐒 , 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Faraut and Korányi (1994, pp. 357–358)).
    8: 28.34 Methods of Computation
  • (d)

    Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • 9: 3.5 Quadrature
    For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004). … For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … Below we give for the classical orthogonal polynomials the recurrence coefficients α n and β n in (3.5.30). … Complex orthogonal polynomials p n ( 1 / ζ ) of degree n = 0 , 1 , 2 , , in 1 / ζ that satisfy the orthogonality condition … …
    10: Bibliography K
  • E. G. Kalnins and W. Miller (1993) Orthogonal Polynomials on n -spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.
  • W. Koepf (1999) Orthogonal polynomials and computer algebra. In Recent developments in complex analysis and computer algebra (Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.), Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.
  • T. H. Koornwinder (1975c) Two-variable Analogues of the Classical Orthogonal Polynomials. In Theory and Application of Special Functions, R. A. Askey (Ed.), pp. 435–495.
  • T. H. Koornwinder (2006) Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.
  • T. Kriecherbauer and K. T.-R. McLaughlin (1999) Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.