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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: 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

… ►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. …

5: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

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External Links:

ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)

Referenced by:

§18.24.

Encodings:

BibTeX

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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Higher-Order SUSY, Exactly Solvable Potentials, and Exceptional Orthogonal Polynomials. Modern Physics Letters A 26, pp. 1843–1852.

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External Links:

Document, Link

Referenced by:

§18.38(iii).

Encodings:

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6: 3.11 Approximation Techniques

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§3.11(i) Minimax Polynomial Approximations

… ►Then there exists a unique $n$th degree polynomial $p_n(x)$, called the minimax (or best uniform) polynomial approximation to $f(x)$ on $[a,b]$, that minimizes ${\max }_{a\le x\le b}\left|{ϵ}_n(x)\right|$, where ${ϵ}_n(x)=f(x)-p_n(x)$. … ►If we have a sufficiently close approximation … ►

§3.11(iii) Minimax Rational Approximations

… ►Then the minimax (or best uniform) rational approximation …

7: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

8: Browsers

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9: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

§18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

►Ismail (1986) gives asymptotic expansions as $n\to \mathrm{\infty }$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_n(\cos \theta ;a,b,c,d|q)$ 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 $q$-Laguerre and continuous $q^{-1}$-Hermite polynomials see Chen and Ismail (1998).

10: 18.24 Hahn Class: Asymptotic Approximations

§18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … ►Corresponding approximations are included for the zeros of $P_n^{(\lambda )}(nx;\varphi )$. ►

Approximations in Terms of Laguerre Polynomials

… ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.