continuous q-ultraspherical polynomials
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§31.5 Solutions Analytic at Three Singularities: Heun Polynomials… ►
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. …
§35.4 Partitions and Zonal Polynomials… ►
Orthogonal Invariance… ►
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). …
§18.3 Definitions►Table 18.3.1 provides the definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and normalization (§§18.2(i) and 18.2(iii)). … ►For exact values of the coefficients of the Jacobi polynomials , the ultraspherical polynomials , the Chebyshev polynomials and , the Legendre polynomials , the Laguerre polynomials , and the Hermite polynomials , see Abramowitz and Stegun (1964, pp. 793–801). … ►For another version of the discrete orthogonality property of the polynomials see (3.11.9). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
§18.28(v) Continuous -Ultraspherical Polynomials… ►These polynomials are also called Rogers polynomials. ►
§18.28(vi) Continuous -Hermite Polynomials… ►
§18.28(vii) Continuous -Hermite Polynomials… ►For continuous -Hermite polynomials the orthogonality measure is not unique. …
§18.30 Associated OP’s►In the recurrence relation (18.2.8) assume that the coefficients , , and are defined when is a continuous nonnegative real variable, and let be an arbitrary positive constant. … ► ►
Associated Jacobi Polynomials… ►
Associated Legendre Polynomials…
Continuous Hahn: .
Continuous Dual Hahn: .
Continuous -Ultraspherical: .
Continuous -Hermite: .
§18.29 Asymptotic Approximations for -Hahn and Askey–Wilson Classes►Ismail (1986) gives asymptotic expansions as , with and other parameters fixed, for continuous -ultraspherical, big and little -Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson the leading term is given by … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). ►For asymptotic approximations to the largest zeros of the -Laguerre and continuous -Hermite polynomials see Chen and Ismail (1998).
§18.21 Hahn Class: Interrelations►
§18.21(i) Dualities… ►
§18.21(ii) Limit Relations and Special Cases… ►
Hahn Jacobi… ►
Continuous Hahn Meixner–Pollaczek…