# Wilson polynomials

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##### 1: 18.26 Wilson Class: Continued
###### §18.26(i) Representations as Generalized Hypergeometric Functions
18.26.5 $\lim_{d\to\infty}\frac{W_{n}\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_% {n}\left(x;a,b,c\right).$
18.26.14 $\ifrac{\delta_{y}\left(W_{n}\left(y^{2};a,b,c,d\right)\right)}{\delta_{y}(y^{2% })}=-n(n+a+b+c+d-1)\*W_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac% {1}{2},d+\tfrac{1}{2}\right).$
###### §18.26(iv) Generating Functions
Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …
##### 2: 18.25 Wilson Class: Definitions
###### §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 $R_{n}\left(x;\gamma,\delta,N\right)$.
##### 3: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
###### §18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as $n\to\infty$, with $x$ and other parameters fixed, for continuous $q$-ultraspherical, big and little $q$-Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson $p_{n}\left(\cos\theta;a,b,c,d\,|\,q\right)$ the leading term is given by …
##### 4: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
###### Wilson Class OP’s
• Wilson: $W_{n}\left(x;a,b,c,d\right)$.

• Askey–Wilson: $p_{n}\left(x;a,b,c,d\,|\,q\right)$.

• ##### 5: 18.28 Askey–Wilson Class
The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of $q$-Racah polynomials, and cases of these families obtained by specialization of parameters. The Askey–Wilson polynomials form a system of OP’s $\{p_{n}(x)\}$, $n=0,1,2,\dots$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. …Both the Askey–Wilson polynomials and the $q$-Racah polynomials can best be described as functions of $z$ (resp. …
For $\omega_{y}$ and $h_{n}$ see Koekoek et al. (2010, Eq. (14.2.2)).
• M. E. H. Ismail (1986) Asymptotics of the Askey-Wilson and $q$-Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.