continuous%20Hahn%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: 18.19 Hahn Class: Definitions
§18.19 Hahn Class: Definitions
… ►Hahn class (or linear lattice class). These are OP’s where the role of is played by or or (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.
Continuous Hahn
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Continuous Hahn: .
Continuous Dual Hahn: .
Continuous -Ultraspherical: .
Continuous -Hermite: .
Continuous -Hermite:
7: 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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§18.25(ii) Weights and Standardizations: Continuous Cases
… ►Continuous Dual Hahn
… ►Table 18.25.2 provides the leading coefficients (§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …8: 18.26 Wilson Class: Continued
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