semi-classical orthogonal polynomials
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1: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
… ►Normalization
… ►Orthogonal Invariance
… ►Summation
… ►Mean-Value
…2: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials
§31.5 Solutions Analytic at Three Singularities: Heun Polynomials
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31.5.2
►is a polynomial of degree , and hence a solution of (31.2.1) that is analytic at all three finite singularities .
These solutions are the Heun polynomials.
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3: 24.18 Physical Applications
§24.18 Physical Applications
►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).4: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …5: 18.3 Definitions
§18.3 Definitions
… ►As given by a Rodrigues formula (18.5.5).
6: 18.2 General Orthogonal Polynomials
§18.2 General Orthogonal Polynomials
… ► … ► … ►Then the OP’s are called semi-classical and (18.2.44) is called a structure relation. …7: Bibliography M
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Symmetric Functions and Orthogonal Polynomials.
University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.
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Orthogonal polynomials associated with root systems.
Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).
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Affine Hecke Algebras and Orthogonal Polynomials.
Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.
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Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.
J. Comput. Appl. Math. 57 (1-2), pp. 215–237.
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Exceptional orthogonal polynomials.
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8: 18.37 Classical OP’s in Two or More Variables
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Orthogonality
… ►The following three conditions, taken together, determine uniquely: … ►Definition in Terms of Jacobi Polynomials
… ►Orthogonality
… ►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …9: René F. Swarttouw
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►Swarttouw is mainly a teacher of mathematics and has published a few papers on special functions and orthogonal polynomials.
He is coauthor of the book Hypergeometric Orthogonal Polynomials and Their
-Analogues) Hypergeometric Orthogonal Polynomials and Their -Analogues.
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