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

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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: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

… ►The Hahn class consists of four discrete families (Hahn, Krawtchouk, Meixner, and Charlier) and two continuous families (continuous Hahn and Meixner–Pollaczek). ►

Hahn, Krawtchouk, Meixner, and Charlier

►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_n(x;\alpha ,\beta ,N)$, Krawtchouk polynomials $K_n(x;p,N)$, Meixner polynomials $M_n(x;\beta ,c)$, and Charlier polynomials $C_n(x;a)$. … ► …

6: Bibliography Q

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

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External Links:

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Referenced by:

§18.38(iii).

Encodings:

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7: 18.21 Hahn Class: Interrelations

§18.21 Hahn Class: Interrelations

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§18.21(i) Dualities

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§18.21(ii) Limit Relations and Special Cases

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Hahn $\to$ Krawtchouk

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Krawtchouk $\to$ Charlier

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8: 18.24 Hahn Class: Asymptotic Approximations

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Krawtchouk

►With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. … ►With $\mu =N/n$ and $x$ fixed, Qiu and Wong (2004) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\mu \in [1,\mathrm{\infty })$. …Asymptotic approximations are also provided for the zeros of $K_n(x;p,N)$ in various cases depending on the values of $p$ and $\mu$. … ►Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.

9: 18.22 Hahn Class: Recurrence Relations and Differences

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§18.22(i) Recurrence Relations in $n$

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Krawtchouk, Meixner, and Charlier

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Krawtchouk, Meixner, and Charlier

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§18.22(iii) $x$-Differences

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Krawtchouk

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10: 18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

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Hahn, Krawtchouk, Meixner, and Charlier

… ►For the Hahn polynomials $p_n(x)=Q_n(x;\alpha ,\beta ,N)$ and …For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and ${\kappa }_n$ are as in Table 18.20.1. … ►

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

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