Krawtchouk polynomials

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1: 18.24 Hahn Class: Asymptotic Approximations

… ►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.

2: 18.21 Hahn Class: Interrelations

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$K_n(x;p,N)=K_x(n;p,N),$ $n,x=0,1,\mathrm{\dots },N$.

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18.21.3 $$\underset{t\to \mathrm{\infty }}{lim}{Q}_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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18.21.6 $$\underset{N\to \mathrm{\infty }}{lim}{K}_n(x;N^{-1}a,N)=C_n(x;a).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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3: 18.19 Hahn Class: Definitions

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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 $C_n(x;a)$. ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

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	$p_n(x)$	$X$	$w_x$	$h_n$
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Krawtchouk	$K_n(x;p,N)$, $n=0,1,\mathrm{\dots },N$	$\{0,1,\mathrm{\dots },N\}$	$\left({\displaystyle \genfrac{}{}{0pt}{}{N}{x}}\right)p^x{(1-p)}^{N-x}$,	${\left({\displaystyle \frac{1-p}{p}}\right)}^n/\left({\displaystyle \genfrac{}{}{0pt}{}{N}{n}}\right)$
…

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Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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$K_n(x;p,N)$	$p^{-n}/{\left(-N\right)}_n$
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4: 18.22 Hahn Class: Recurrence Relations and Differences

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Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_n(x)$	$A_n$	$C_n$
$K_n(x;p,N)$	$p(N-n)$	$n(1-p)$
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Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_n(x)$	$A(x)$	$C(x)$	${\lambda }_n$
$K_n(x;p,N)$	$p(x-N)$	$(p-1)x$	$-n$
…

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18.22.21 $${\mathrm{\Delta }}_x{K}_n(x;p,N)=-\frac{n}{pN}{K}_{n-1}(x;p,N-1),$$

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\mathrm{\Delta }$: forward difference operator, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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18.22.22 $${\nabla }_x\left(\left(\genfrac{}{}{0.0pt}{}{N}{x}\right)p^x{(1-p)}^{N-x}{K}_n(x;p,N)\right)=\left(\genfrac{}{}{0.0pt}{}{N+1}{x}\right)p^x{(1-p)}^{N-x}{K}_{n+1}(x;p,N+1).$$

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, ${\nabla }_x$: backward difference, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.22.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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5: 18.20 Hahn Class: Explicit Representations

… ►For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and ${\kappa }_n$ are as in Table 18.20.1. ►

Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).

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$p_n(x)$	$F(x)$	${\kappa }_n$
$K_n(x;p,N)$	$x-N$	${\left(-N\right)}_n$
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18.20.6 $$K_n(x;p,N)={}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,-x}{-N};p^{-1}),$$ $n=0,1,\mathrm{\dots },N$.

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19, §18.21(ii)

Permalink:

http://dlmf.nist.gov/18.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(ii), §18.20 and Ch.18

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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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7: 18.23 Hahn Class: Generating Functions

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18.23.3 $${\left(1-\frac{1-p}{p}z\right)}^x{(1+z)}^{N-x}=\sum _{n=0}^N\left(\genfrac{}{}{0.0pt}{}{N}{n}\right)K_n(x;p,N)z^n,$$ $x=0,1,\mathrm{\dots },N$.

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $N$: positive integer, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E3

Encodings:

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See also:

Annotations for §18.23, §18.23 and Ch.18

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8: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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Krawtchouk: $K_n(x;p,N)$.

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9: 18.38 Mathematical Applications

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Coding Theory

►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: 18.26 Wilson Class: Continued

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18.26.11 $$\underset{t\to \mathrm{\infty }}{lim}{R}_n(x(x+t+1);pt,(1-p)t,N)=K_n(x;p,N).$$

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Symbols:

$K_n(x;p,N)$: Krawtchouk polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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