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

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Defines:

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

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

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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: 18.25 Wilson Class: Definitions

… ►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 $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►

Continuous Dual Hahn

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Dual Hahn

… ►Table 18.25.2 provides the leading coefficients $k_n$ (§18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …

4: 18.26 Wilson Class: Continued

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Wilson $\to$ Continuous Dual Hahn

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Wilson $\to$ Continuous Hahn

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Continuous Dual Hahn $\to$ Meixner–Pollaczek

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Continuous Dual Hahn

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Dual Hahn

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5: 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). …

6: 18.1 Notation

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Continuous Hahn: $p_n(x;a,b,\overline{a},\overline{b})$.

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Continuous Dual Hahn: $S_n(x;a,b,c)$.

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Continuous $q$-Ultraspherical: $C_n(x;\beta |q)$.

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Continuous $q$-Hermite: $H_n\left(x|q\right)$.

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Continuous $q^{-1}$-Hermite: $h_n\left(x|q\right)$

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7: 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 $n=12$ for these polynomials and for coefficients up to $n=6$ 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. …

8: 20 Theta Functions

Chapter 20 Theta Functions

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9: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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Duality of Hahn and Dual Hahn

… ►For the dual Hahn polynomial $R_n(x;\gamma ,\delta ,N)$ see §18.25. … ►

§18.21(ii) Limit Relations and Special Cases

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Continuous Hahn $\to$ Meixner–Pollaczek

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10: 6.16 Mathematical Applications

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … … ►

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify