Racah%20polynomials
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1: 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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2: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
… ►Normalization
… ►Orthogonal Invariance
… ►Summation
… ►Mean-Value
…3: 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). …4: 18.3 Definitions
§18.3 Definitions
… ►For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). …For explicit power series coefficients up to for these polynomials and for coefficients up to for Jacobi and ultraspherical polynomials see Abramowitz and Stegun (1964, pp. 793–801). … ►Bessel polynomials
►Bessel polynomials are often included among the classical OP’s. …5: 18.26 Wilson Class: Continued
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§18.26(ii) Limit Relations
… ►Racah Dual Hahn
… ►Racah Hahn
… ►Racah
… ►Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …6: 18.25 Wilson Class: Definitions
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►The Wilson class consists of two discrete families (Racah and dual Hahn) and two continuous families (Wilson and continuous dual Hahn).
►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials
, continuous dual Hahn polynomials
, Racah polynomials
, and dual Hahn polynomials
.
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Further Constraints for Racah Polynomials
… ►Racah
… ►Table 18.25.2 provides the leading coefficients (§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …7: 18.1 Notation
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►Nor do we consider the shifted Jacobi polynomials:
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Classical OP’s
… ►Hahn Class OP’s
… ►Racah: .
-Racah: .
8: 18.38 Mathematical Applications
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Coding Theory
►For applications of Krawtchouk polynomials and -Racah polynomials to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987). … ►The symbol (34.4.3), with an alternative expression as a terminating balanced of unit argument, can be expressend in terms of Racah polynomials (18.26.3). The orthogonality relations (34.5.14) for the symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … …9: 18.28 Askey–Wilson Class
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►Both the Askey–Wilson polynomials and the -Racah polynomials can best be described as functions of (resp.
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