Jacobi matrix
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1: 35.7 Gaussian Hypergeometric Function of Matrix Argument
2: 18.2 General Orthogonal Polynomials
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►The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix.
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►When the Jacobi matrix in (18.2.11_9) is truncated to an
matrix
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3: Errata
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Chapter 35 Functions of Matrix Argument
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The generalized hypergeometric function of matrix argument , was linked inadvertently as its single variable counterpart . Furthermore, the Jacobi function of matrix argument , and the Laguerre function of matrix argument , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by , and . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.
4: 3.2 Linear Algebra
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►Also, the recurrence relations in (3.2.23) and (3.5.30) are similar, as well as the matrix
in (3.2.22) and the Jacobi matrix
in (3.5.31).
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5: 3.5 Quadrature
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►The Gauss nodes (the zeros of ) are the eigenvalues of the (symmetric tridiagonal) Jacobi matrix of order :
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6: 15.17 Mathematical Applications
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§15.17(iii) Group Representations
►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL, and spherical functions on certain nonsymmetric Gelfand pairs. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …7: 29.15 Fourier Series and Chebyshev Series
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►be the tridiagonal matrix with , , as in (29.3.11), (29.3.12).
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►Since (29.2.5) implies that , (29.15.1) can be rewritten in the form
…This determines the polynomial of degree for which ; compare Table 29.12.1.
The set of coefficients of this polynomial (without normalization) can also be found directly as an eigenvector of an tridiagonal matrix; see Arscott and Khabaza (1962).
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29.15.49
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8: Bibliography K
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Orthogonal Polynomials on -spheres: Gegenbauer, Jacobi and Heun.
In Topics in Polynomials of One and Several Variables and their
Applications,
pp. 299–322.
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Cyclic identities for Jacobi elliptic and related functions.
J. Math. Phys. 44 (4), pp. 1822–1841.
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Connecting Jacobi elliptic functions with different modulus parameters.
Pramana 63 (5), pp. 921–936.
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The efficient evaluation of the hypergeometric function of a matrix argument.
Math. Comp. 75 (254), pp. 833–846.
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Hypergeometric functions of
matrix argument are expressible in terms of Appel’s functions
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Proc. Amer. Math. Soc. 70 (1), pp. 39–42.
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9: Bibliography I
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The -matrix method.
Adv. in Appl. Math. 46 (1-4), pp. 379–395.
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Two families of orthogonal polynomials related to Jacobi polynomials.
Rocky Mountain J. Math. 21 (1), pp. 359–375.
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Asymptotics of the Askey-Wilson and -Jacobi polynomials.
SIAM J. Math. Anal. 17 (6), pp. 1475–1482.
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10: 18.38 Mathematical Applications
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►The Askey–Gasper inequality
…also the case of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane.
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