# §18.21 Hahn Class: Interrelations

## §18.21(i) Dualities

### Duality of Hahn and Dual Hahn

 18.21.1 $Q_{n}\left(x;\alpha,\beta,N\right)=R_{x}\left(n(n+\alpha+\beta+1);\alpha,\beta% ,N\right),$ $n,x=0,1,\dots,N$.

For the dual Hahn polynomial $R_{n}\left(x;\gamma,\delta,N\right)$ see §18.25.

### Self-Dualities

 18.21.2 $\displaystyle K_{n}\left(x;p,N\right)$ $\displaystyle=K_{x}\left(n;p,N\right),$ $n,x=0,1,\dots,N$. $\displaystyle M_{n}\left(x;\beta,c\right)$ $\displaystyle=M_{x}\left(n;\beta,c\right),$ $n,x=0,1,2,\dots$. $\displaystyle C_{n}\left(x;a\right)$ $\displaystyle=C_{x}\left(n;a\right),$ $n,x=0,1,2,\dots$.

## §18.21(ii) Limit Relations and Special Cases

### Hahn $\to$ Krawtchouk

 18.21.3 $\lim_{t\to\infty}Q_{n}\left(x;pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$

### Hahn $\to$ Meixner

 18.21.4 $\lim_{N\to\infty}Q_{n}\left(x;b-1,N(c^{-1}-1),N\right)=M_{n}\left(x;b,c\right).$

### Hahn $\to$ Jacobi

 18.21.5 $\lim_{N\to\infty}Q_{n}\left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_% {n}\left(1-2x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}.$

### Krawtchouk $\to$ Charlier

 18.21.6 $\lim_{N\to\infty}K_{n}\left(x;N^{-1}a,N\right)=C_{n}\left(x;a\right).$

### Meixner $\to$ Charlier

 18.21.7 $\lim_{\beta\to\infty}M_{n}\left(x;\beta,a(a+\beta)^{-1}\right)=C_{n}\left(x;a% \right).$ ⓘ 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: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

### Meixner $\to$ Laguerre

 18.21.8 $\lim_{c\to 1}M_{n}\left((1-c)^{-1}x;\alpha+1,c\right)=\frac{L^{(\alpha)}_{n}% \left(x\right)}{L^{(\alpha)}_{n}\left(0\right)}.$

### Charlier $\to$ Hermite

 18.21.9 $\lim_{a\to\infty}(2a)^{\frac{1}{2}n}C_{n}\left((2a)^{\frac{1}{2}}x+a;a\right)=% (-1)^{n}H_{n}\left(x\right).$ ⓘ 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: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

### Continuous Hahn $\to$ Meixner–Pollaczek

 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 $p_{n}\left(x;a,a+\tfrac{1}{2},a,a+\tfrac{1}{2}\right)=2^{-2n}{\left(4a+n\right% )_{n}}P^{(2a)}_{n}\left(2x;\tfrac{1}{2}\pi\right).$

### Meixner–Pollaczek $\to$ Laguerre

 18.21.12 $\lim_{\phi\to 0}P^{(\frac{1}{2}\alpha+\frac{1}{2})}_{n}\left(-(2\phi)^{-1}x;% \phi\right)=L^{(\alpha)}_{n}\left(x\right).$

A graphical representation of limits in §§18.7(iii), 18.21(ii), and 18.26(ii) is provided by the Askey scheme depicted in Figure 18.21.1.