Charlier polynomials

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1—10 of 12 matching pages

1: 18.21 Hahn Class: Interrelations

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C_{n}\left(x;a\right)=C_{x}\left(n;a\right),

n,x=0,1,2,\dots

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18.21.6

\lim_{N\to\infty}K_{n}\left(x;N^{-1}a,N\right)=C_{n}\left(x;a\right).

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : 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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18.21.7

\lim_{\beta\to\infty}M_{n}\left(x;\beta,a(a+\beta)^{-1}\right)=C_{n}\left(x;a% \right).

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii), §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.21.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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Charlier $\to$ Hermite

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18.21.9

\lim_{a\to\infty}(2a)^{\frac{1}{2}n}C_{n}\left((2a)^{\frac{1}{2}}x+a;a\right)=% (-1)^{n}H_{n}\left(x\right).

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.21.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

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2: 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}\left(x;\alpha,\beta,N\right)

, Krawtchouk polynomials

K_{n}\left(x;p,N\right)

, Meixner polynomials

M_{n}\left(x;\beta,c\right)

, and Charlier polynomials

C_{n}\left(x;a\right)

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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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	$p_{n}(x)$	$X$	$w_{x}$	$h_{n}$
…
Charlier	$C_{n}\left(x;a\right)$	$\{0,1,2,\dots\}$	$\ifrac{a^{x}}{x!}$ , $a>0$	$a^{-n}{\mathrm{e}}^{a}n!$

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

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$p_{n}(x)$	$k_{n}$
…
$C_{n}\left(x;a\right)$	$(-a)^{-n}$

► …

3: 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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$p_{n}(x)$	$A_{n}$	$C_{n}$
…
$C_{n}\left(x;a\right)$	$a$	$n$

► … ►

Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_{n}(x)$	$A(x)$	$C(x)$	$\lambda_{n}$
…
$C_{n}\left(x;a\right)$	$a$	$x$	$n$

► … ►

18.22.25

\Delta_{x}C_{n}\left(x;a\right)=-\frac{n}{a}C_{n-1}\left(x;a\right),

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\Delta$ : forward difference operator, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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18.22.26

\nabla_{x}\left(\frac{a^{x}}{x!}C_{n}\left(x;a\right)\right)=\frac{a^{x}}{x!}C% _{n+1}\left(x;a\right).

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\nabla_{\NVar{x}}$ : backward difference, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

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4: 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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$p_{n}(x)$	$F(x)$	$\kappa_{n}$
…
$C_{n}\left(x;a\right)$	$1$	$1$

► … ►

18.20.8

C_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,-x\atop-};-a^{-1}\right).

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §13.6(v), §18.19, §18.20(ii), §18.21(ii)
Permalink:: http://dlmf.nist.gov/18.20.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §18.20(ii), §18.20 and Ch.18

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5: 18.23 Hahn Class: Generating Functions

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18.23.5

{\mathrm{e}}^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}% \left(x;a\right)}{n!}z^{n},

x=0,1,2,\dots

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.2(xii), §18.23
Permalink:: http://dlmf.nist.gov/18.23.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.23, §18.23 and Ch.18

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6: 18.24 Hahn Class: Asymptotic Approximations

… ►Dunster (2001b) provides various asymptotic expansions for

C_{n}\left(x;a\right)

n\to\infty

, in terms of elementary functions or in terms of Bessel functions. …

7: 13.6 Relations to Other Functions

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Charlier Polynomials

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13.6.20

U\left(-n,z-n+1,a\right)={\left(-z\right)_{n}}M\left(-n,z-n+1,a\right)=a^{n}C_% {n}\left(z;a\right).

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Symbols:: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.6(v)
Permalink:: http://dlmf.nist.gov/13.6.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

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8: 18.1 Notation

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\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.

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Charlier: $C_{n}\left(x;a\right)$ .

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9: Bibliography D

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T. M. Dunster (2001b) Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.

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External Links:: ISSN 0021-9045, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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10: Bibliography G

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W. M. Y. Goh (1998) Plancherel-Rotach asymptotics for the Charlier polynomials. Constr. Approx. 14 (2), pp. 151–168.

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External Links:: ISSN 0176-4276, Document, MathReview (Timothy S. Norfolk), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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Charlier polynomials

1—10 of 12 matching pages

1: 18.21 Hahn Class: Interrelations

Charlier → Hermite

2: 18.19 Hahn Class: Definitions

Hahn, Krawtchouk, Meixner, and Charlier

3: 18.22 Hahn Class: Recurrence Relations and Differences

4: 18.20 Hahn Class: Explicit Representations

5: 18.23 Hahn Class: Generating Functions

6: 18.24 Hahn Class: Asymptotic Approximations

7: 13.6 Relations to Other Functions

Charlier Polynomials

8: 18.1 Notation

9: Bibliography D

10: Bibliography G

Charlier $\to$ Hermite