# Meixner polynomials

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##### 1: 18.24 Hahn Class: Asymptotic Approximations
For two asymptotic expansions of $M_{n}\left(nx;\beta,c\right)$ as $n\to\infty$, with $\beta$ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). … For asymptotic approximations for the zeros of $M_{n}\left(nx;\beta,c\right)$ in terms of zeros of $\operatorname{Ai}\left(x\right)$9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … For an asymptotic expansion of $P^{(\lambda)}_{n}\left(nx;\phi\right)$ as $n\to\infty$, with $\phi$ fixed, see Li and Wong (2001). …Corresponding approximations are included for the zeros of $P^{(\lambda)}_{n}\left(nx;\phi\right)$. … For asymptotic approximations to $P^{(\lambda)}_{n}\left(x;\phi\right)$ as $|x+i\lambda|\to\infty$, with $n$ fixed, see Temme and López (2001). …
##### 2: 18.21 Hahn Class: Interrelations
$M_{n}\left(x;\beta,c\right)=M_{x}\left(n;\beta,c\right),$ $n,x=0,1,2,\dots$.
18.21.10 $\lim_{t\to\infty}t^{-n}p_{n}\left(x-t;\lambda+it,-t\tan\phi,\lambda-it,-t\tan% \phi\right)=\frac{(-1)^{n}}{(\cos\phi)^{n}}P^{(\lambda)}_{n}\left(x;\phi\right).$
18.21.11 $p_{n}\left(x;a,a+\tfrac{1}{2},a,a+\tfrac{1}{2}\right)=2^{-2n}{\left(4a+n\right% )_{n}}P^{(2a)}_{n}\left(2x;\tfrac{1}{2}\pi\right).$
18.21.12 $\lim_{\phi\to 0}P^{(\frac{1}{2}\alpha+\frac{1}{2})}_{n}\left(-(2\phi)^{-1}x;% \phi\right)=L^{(\alpha)}_{n}\left(x\right).$
##### 3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.29 $\delta_{x}\left(P^{(\lambda)}_{n}\left(x;\phi\right)\right)=2\sin\phi P^{(% \lambda+\frac{1}{2})}_{n-1}\left(x;\phi\right),$
##### 4: 18.19 Hahn Class: Definitions
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 $C_{n}\left(x;a\right)$.
##### 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.
18.20.4 $w^{(\lambda)}(x;\phi)P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{1}{n!}\delta_{% x}^{n}\left(w^{(\lambda+\frac{1}{2}n)}(x;\phi)\right).$
18.20.7 $M_{n}\left(x;\beta,c\right)={{}_{2}F_{1}}\left({-n,-x\atop\beta};1-c^{-1}% \right).$
18.20.10 $P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}e^{% \mathrm{i}n\phi}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm{i}x\atop 2\lambda};1-% e^{-2\mathrm{i}\phi}\right).$
##### 6: 18.23 Hahn Class: Generating Functions
18.23.4 $\left(1-\frac{z}{c}\right)^{x}(1-z)^{-x-\beta}=\sum_{n=0}^{\infty}\frac{{\left% (\beta\right)_{n}}}{n!}M_{n}\left(x;\beta,c\right)z^{n},$ $x=0,1,2,\dots$, $|z|<1$.
18.23.7 $(1-e^{\mathrm{i}\phi}z)^{-\lambda+\mathrm{i}x}(1-e^{-\mathrm{i}\phi}z)^{-% \lambda-\mathrm{i}x}=\sum_{n=0}^{\infty}P^{(\lambda)}_{n}\left(x;\phi\right)z^% {n},$ $|z|<1$.
##### 7: 18.35 Pollaczek Polynomials
18.35.9 $P^{(\lambda)}_{n}\left(\cos\phi;0,x\sin\phi\right)=P^{(\lambda)}_{n}\left(x;% \phi\right).$
For the polynomials $C^{(\lambda)}_{n}\left(x\right)$ and $P^{(\lambda)}_{n}\left(x;\phi\right)$ see §§18.3 and 18.19, respectively. …
##### 8: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
• Meixner: $M_{n}\left(x;\beta,c\right)$.

• Meixner–Pollaczek: $P^{(\lambda)}_{n}\left(x;\phi\right)$.

• ##### 9: 18.26 Wilson Class: Continued
18.26.8 $\lim_{t\to\infty}\ifrac{S_{n}\left((x-t)^{2};\lambda+it,\lambda-it,t\cot\phi% \right)}{t^{n}}=n!(\csc\phi)^{n}P^{(\lambda)}_{n}\left(x;\phi\right).$
18.26.13 $\lim_{N\to\infty}R_{n}\left(r(x;\beta,c,N);\beta-1,c^{-1}(1-c)N,N\right)=M_{n}% \left(x;\beta,c\right).$
##### 10: 15.9 Relations to Other Functions
###### Meixner
15.9.9 $M_{n}\left(x;\beta,c\right)=F\left({-n,-x\atop\beta};1-\frac{1}{c}\right).$
###### Meixner–Pollaczek
15.9.10 $P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{% \mathrm{e}}^{n\mathrm{i}\phi}F\left({-n,\lambda+\mathrm{i}x\atop 2\lambda};1-{% \mathrm{e}}^{-2\mathrm{i}\phi}\right).$