# §18.21 Hahn Class: Interrelations

## §18.21(i) Dualities

### Duality of Hahn and Dual Hahn

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

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

### Self-Dualities

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

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

### Hahn $\to$ Krawtchouk

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

### Hahn $\to$ Meixner

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

### Hahn $\to$ Jacobi

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

### Krawtchouk $\to$ Charlier

 18.21.6 $\lim_{N\to\infty}\mathop{K_{n}\/}\nolimits\!\left(x;N^{-1}a,N\right)=\mathop{C% _{n}\/}\nolimits\!\left(x;a\right).$

### Meixner $\to$ Charlier

 18.21.7 $\lim_{\beta\to\infty}\mathop{M_{n}\/}\nolimits\!\left(x;\beta,a(a+\beta)^{-1}% \right)=\mathop{C_{n}\/}\nolimits\!\left(x;a\right).$

### Meixner $\to$ Laguerre

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

### Charlier $\to$ Hermite

 18.21.9 $\lim_{a\to\infty}(2a)^{\frac{1}{2}n}\mathop{C_{n}\/}\nolimits\!\left((2a)^{% \frac{1}{2}}x+a;a\right)=(-1)^{n}\mathop{H_{n}\/}\nolimits\!\left(x\right).$

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

 18.21.10 $\lim_{t\to\infty}t^{-n}\mathop{p_{n}\/}\nolimits\!\left(x-t;\lambda+it,-t% \mathop{\tan\/}\nolimits\phi,\lambda-it,-t\mathop{\tan\/}\nolimits\phi\right)=% \frac{(-1)^{n}}{(\mathop{\cos\/}\nolimits\phi)^{n}}\mathop{P^{(\lambda)}_{n}\/% }\nolimits\!\left(x;\phi\right).$
 18.21.11 $\mathop{p_{n}\/}\nolimits\!\left(x;a,a+\tfrac{1}{2},a,a+\tfrac{1}{2}\right)=2^% {-2n}{\left(4a+n\right)_{n}}\mathop{P^{(2a)}_{n}\/}\nolimits\!\left(2x;\tfrac{% 1}{2}\pi\right).$

### Meixner–Pollaczek $\to$ Laguerre

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

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.