# Charlier polynomials

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## 1—10 of 11 matching pages

##### 1: 18.21 Hahn Class: Interrelations
$C_{n}\left(x;a\right)=C_{x}\left(n;a\right),$ $n,x=0,1,2,\dots$.
##### 2: 18.19 Hahn Class: Definitions
###### Hahn, Krawtchouk, Meixner, and Charlier
Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§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 $C_{n}\left(x;a\right)$.
##### 3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.26 $\nabla_{x}\left(\frac{a^{x}}{x!}C_{n}\left(x;a\right)\right)=\frac{a^{x}}{x!}C% _{n+1}\left(x;a\right).$
##### 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.
18.20.8 $C_{n}\left(x;a\right)={{}_{2}F_{0}}\left({-n,-x\atop-};-a^{-1}\right).$
##### 5: 18.23 Hahn Class: Generating Functions
18.23.5 $e^{z}\left(1-\frac{z}{a}\right)^{x}=\sum_{n=0}^{\infty}\frac{C_{n}\left(x;a% \right)}{n!}z^{n},$ $x=0,1,2,\dots$.
##### 6: 18.24 Hahn Class: Asymptotic Approximations
Dunster (2001b) provides various asymptotic expansions for $C_{n}\left(x;a\right)$ as $n\to\infty$, in terms of elementary functions or in terms of Bessel functions. …
##### 7: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
• Charlier: $C_{n}\left(x;a\right)$.

• ##### 8: 13.6 Relations to Other Functions
###### CharlierPolynomials
13.6.20 $U\left(-n,z-n+1,a\right)={\left(-z\right)_{n}}M\left(-n,z-n+1,a\right)=a^{n}C_% {n}\left(z;a\right).$
##### 9: Bibliography D
• T. M. Dunster (2001b) Uniform asymptotic expansions for Charlier polynomials. J. Approx. Theory 112 (1), pp. 93–133.
• ##### 10: Bibliography G
• W. M. Y. Goh (1998) Plancherel-Rotach asymptotics for the Charlier polynomials. Constr. Approx. 14 (2), pp. 151–168.