# Hahn polynomials

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##### 1: 18.21 Hahn Class: Interrelations
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. …
18.21.3 $\lim_{t\to\infty}Q_{n}\left(x;pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$
18.21.4 $\lim_{N\to\infty}Q_{n}\left(x;b-1,N(c^{-1}-1),N\right)=M_{n}\left(x;b,c\right).$
18.21.5 $\lim_{N\to\infty}Q_{n}\left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_% {n}\left(1-2x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}.$
##### 2: 18.19 Hahn Class: Definitions
###### §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 $C_{n}\left(x;a\right)$. …
18.19.1 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
##### 3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.1 $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),$
18.22.4 $q_{n}(x)=\ifrac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{p_{n}\left(% \mathrm{i}a;a,b,\overline{a},\overline{b}\right)},$
18.22.9 $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),$
18.22.13 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
##### 4: 18.23 Hahn Class: Generating Functions
###### Hahn
18.23.1 ${{}_{1}F_{1}}\left({-x\atop\alpha+1};-z\right){{}_{1}F_{1}}\left({x-N\atop% \beta+1};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}}{{\left(\beta+1% \right)_{n}}n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},$ $x=0,1,\dots,N$.
18.23.2 ${{}_{2}F_{0}}\left({-x,-x+\beta+N+1\atop-};-z\right)\*{{}_{2}F_{0}}\left({x-N,% x+\alpha+1\atop-};z\right)=\sum_{n=0}^{N}\frac{{\left(-N\right)_{n}}{\left(% \alpha+1\right)_{n}}}{n!}Q_{n}\left(x;\alpha,\beta,N\right)z^{n},$ $x=0,1,\dots,N$.
18.23.6 ${{}_{1}F_{1}}\left({a+\mathrm{i}x\atop 2\Re a};-\mathrm{i}z\right){{}_{1}F_{1}% }\left({\overline{b}-\mathrm{i}x\atop 2\Re b};\mathrm{i}z\right)=\sum_{n=0}^{% \infty}\frac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{{\left(2\Re a% \right)_{n}}{\left(2\Re b\right)_{n}}}z^{n}.$
##### 5: 18.20 Hahn Class: Explicit Representations
For the Hahn polynomials $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right)$ and …
18.20.3 $w(x;a,b,\overline{a},\overline{b})p_{n}\left(x;a,b,\overline{a},\overline{b}% \right)=\frac{1}{n!}\delta_{x}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac{1}{2}n,% \overline{a}+\tfrac{1}{2}n,\overline{b}+\tfrac{1}{2}n)\right).$
###### §18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions
18.20.5 $Q_{n}\left(x;\alpha,\beta,N\right)={{}_{3}F_{2}}\left({-n,n+\alpha+\beta+1,-x% \atop\alpha+1,-N};1\right),$ $n=0,1,\dots,N$.
(For symmetry properties of $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …
##### 6: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
###### Hahn Class OP’s
• Hahn: $Q_{n}\left(x;\alpha,\beta,N\right)$.

• Continuous Hahn: $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$.

• $q$-Hahn: $Q_{n}\left(x;\alpha,\beta,N;q\right)$.

• ##### 7: 18.26 Wilson Class: Continued
18.26.5 $\lim_{d\to\infty}\frac{W_{n}\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_% {n}\left(x;a,b,c\right).$
18.26.6 $\lim_{t\to\infty}\frac{W_{n}\left((x+t)^{2};a-it,b-it,\overline{a}+it,% \overline{b}+it\right)}{(-2t)^{n}n!}=p_{n}\left(x;a,b,\overline{a},\overline{b% }\right).$
18.26.10 $\lim_{\delta\to\infty}R_{n}\left(x(x+\gamma+\delta+1);\alpha,\beta,-N-1,\delta% \right)=Q_{n}\left(x;\alpha,\beta,N\right).$
18.26.11 $\lim_{t\to\infty}R_{n}\left(x(x+t+1);pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$
18.26.13 $\lim_{N\to\infty}R_{n}\left(r(x;\beta,c,N);\beta-1,c^{-1}(1-c)N,N\right)=M_{n}% \left(x;\beta,c\right).$
##### 8: 18.24 Hahn Class: Asymptotic Approximations
###### §18.24 Hahn Class: Asymptotic Approximations
When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval(0,1), uniform asymptotic formulas (as $n\to\infty$ ) of the Hahn polynomials $Q_{n}(z;\alpha,\beta,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.
##### 9: 18.25 Wilson Class: Definitions
Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_{n}\left(x;a,b,c,d\right)$, continuous dual Hahn polynomials $S_{n}\left(x;a,b,c\right)$, Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$, and dual Hahn polynomials $R_{n}\left(x;\gamma,\delta,N\right)$.
18.25.13 $p_{n}(x)=R_{n}\left(x;\gamma,\delta,N\right),$
Table 18.25.2 provides the leading coefficients $k_{n}$18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
##### 10: 18.27 $q$-Hahn Class
The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. …
###### §18.27(ii) $q$-HahnPolynomials
18.27.3 $Q_{n}(x)=Q_{n}\left(x;\alpha,\beta,N;q\right)={{}_{3}\phi_{2}}\left({q^{-n},% \alpha\beta q^{n+1},x\atop\alpha q,q^{-N}};q,q\right),$ $n=0,1,\dots,N$.