# continuous dual Hahn polynomials

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## 3 matching pages

##### 1: 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. …
##### 2: 18.26 Wilson Class: Continued
18.26.2 $\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{% n}}}={{}_{3}F_{2}}\left({-n,a+iy,a-iy\atop a+b,a+c};1\right).$
18.26.5 $\lim_{d\to\infty}\frac{W_{n}\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_% {n}\left(x;a,b,c\right).$
18.26.8 $\lim_{t\to\infty}\ifrac{S_{n}\left((x-t)^{2};\lambda+it,\lambda-it,t\cot\phi% \right)}{t^{n}}=n!(\csc\phi)^{n}P^{(\lambda)}_{n}\left(x;\phi\right).$
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 $(1-z)^{-c+iy}{{}_{2}F_{1}}\left({a+iy,b+iy\atop a+b};z\right)=\sum_{n=0}^{% \infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b\right)_{n}}n!}z^{n},$ $|z|<1$.
##### 3: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$