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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 $${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

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See also:

Annotations for §31.5 and Ch.31

►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. …

2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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Normalization

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Orthogonal Invariance

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Summation

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Mean-Value

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3: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

4: 18.3 Definitions

§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 $P_n^{(\alpha ,\beta )}\left(x\right)$, the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Chebyshev polynomials $T_n\left(x\right)$ and $U_n\left(x\right)$, the Legendre polynomials $P_n\left(x\right)$, the Laguerre polynomials $L_n\left(x\right)$, and the Hermite polynomials $H_n\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … ►For another version of the discrete orthogonality property of the polynomials $T_n\left(x\right)$ see (3.11.9). … ►Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …

5: 18 Orthogonal Polynomials

Chapter 18 Orthogonal Polynomials

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6: 18.30 Associated OP’s

§18.30 Associated OP’s

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Associated Jacobi Polynomials

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Associated Legendre Polynomials

… ►For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12); on associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). …

7: 18.28 Askey–Wilson Class

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§18.28(iii) Al-Salam–Chihara Polynomials

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§18.28(iv) $q^{-1}$-Al-Salam–Chihara Polynomials

… ►For further nondegenerate cases see Chihara and Ismail (1993) and Christiansen and Ismail (2006). ►

§18.28(v) Continuous $q$-Ultraspherical Polynomials

… ►These polynomials are also called Rogers polynomials. …

8: 18.1 Notation

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Classical OP’s

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Hahn Class OP’s

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Wilson Class OP’s

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Al-Salam–Chihara: $Q_n(x;a,b|q)$.

… ►Nor do we consider the shifted Jacobi polynomials: …

9: 18.38 Mathematical Applications

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Approximation Theory

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Integrable Systems

… ►Ultraspherical polynomials are zonal spherical harmonics. … ►

Group Representations

… ►For applications of Krawtchouk polynomials $K_n(x;p,N)$ and $q$-Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).

10: 18.2 General Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

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§18.2(ii) $x$-Difference Operators

… ►This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). … ► … ►

§18.2(vi) Zeros

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