# continuous Hahn polynomials

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##### 1: 18.20 Hahn Class: Explicit Representations
18.20.3 $w(x;a,b,\overline{a},\overline{b})p_{n}\left(x;a,b,\overline{a},\overline{b}% \right)=\frac{1}{n!}\delta_{x}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac{1}{2}n,% \overline{a}+\tfrac{1}{2}n,\overline{b}+\tfrac{1}{2}n)\right).$
18.20.9 $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)=\frac{{\mathrm{i}}^{n}{\left% (a+\overline{a}\right)_{n}}{\left(a+\overline{b}\right)_{n}}}{n!}\*{{}_{3}F_{2% }}\left({-n,n+2\Re\left(a+b\right)-1,a+\mathrm{i}x\atop a+\overline{a},a+% \overline{b}};1\right).$
(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).) …
##### 2: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.4 $q_{n}(x)=\ifrac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{p_{n}\left(% \mathrm{i}a;a,b,\overline{a},\overline{b}\right)},$
18.22.13 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
18.22.27 $\delta_{x}\left(p_{n}\left(x;a,b,\overline{a},\overline{b}\right)\right)=(n+2% \Re\left(a+b\right)-1)\*p_{n-1}\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline% {a}+\tfrac{1}{2},\overline{b}+\tfrac{1}{2}\right),$
##### 3: 18.19 Hahn Class: Definitions
18.19.1 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
##### 4: 18.21 Hahn Class: Interrelations
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 $p_{n}\left(x;a,a+\tfrac{1}{2},a,a+\tfrac{1}{2}\right)=2^{-2n}{\left(4a+n\right% )_{n}}P^{(2a)}_{n}\left(2x;\tfrac{1}{2}\pi\right).$ Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify
##### 5: 18.23 Hahn Class: Generating Functions
18.23.6 ${{}_{1}F_{1}}\left({a+\mathrm{i}x\atop 2\Re a};-\mathrm{i}z\right){{}_{1}F_{1}% }\left({\overline{b}-\mathrm{i}x\atop 2\Re b};\mathrm{i}z\right)=\sum_{n=0}^{% \infty}\frac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{{\left(2\Re a% \right)_{n}}{\left(2\Re b\right)_{n}}}z^{n}.$
##### 6: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
• Continuous Hahn: $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$.

• ##### 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 $\lim_{t\to\infty}\frac{W_{n}\left((x+t)^{2};a-it,b-it,\overline{a}+it,% \overline{b}+it\right)}{(-2t)^{n}n!}=p_{n}\left(x;a,b,\overline{a},\overline{b% }\right).$
18.26.8 $\lim_{t\to\infty}\ifrac{S_{n}\left((x-t)^{2};\lambda+it,\lambda-it,t\cot\phi% \right)}{t^{n}}=n!(\csc\phi)^{n}P^{(\lambda)}_{n}\left(x;\phi\right).$
18.26.15 $\ifrac{\delta_{y}\left(S_{n}\left(y^{2};a,b,c\right)\right)}{\delta_{y}(y^{2})% }=-nS_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2}\right).$
18.26.19 $(1-z)^{-c+iy}{{}_{2}F_{1}}\left({a+iy,b+iy\atop a+b};z\right)=\sum_{n=0}^{% \infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b\right)_{n}}n!}z^{n},$ $|z|<1$.
##### 8: 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 $R_{n}\left(x;\gamma,\delta,N\right)$. Table 18.25.2 provides the leading coefficients $k_{n}$18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
##### 9: 18.2 General Orthogonal Polynomials
This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials18.20(i)). …
##### 10: Bibliography
• R. Askey (1985) Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.