# §18.25 Wilson Class: Definitions

## §18.25(i) Preliminaries

For the Wilson class OP’s $p_{n}(x)$ with $x=\lambda(y)$: if the $y$-orthogonality set is $\{0,1,\dots,N\}$, then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ in the Jacobi, Laguerre, and Hermite cases is played by the operator $\Delta_{y}$ followed by division by $\Delta_{y}(\lambda(y))$, or by the operator $\nabla_{y}$ followed by division by $\nabla_{y}(\lambda(y))$. Alternatively if the $y$-orthogonality interval is $(0,\infty)$, then the role of $\ifrac{\mathrm{d}}{\mathrm{d}x}$ is played by the operator $\delta_{y}$ followed by division by $\delta_{y}(\lambda(y))$.

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 $R_{n}\left(x;\gamma,\delta,N\right)$.

### Further Constraints for Racah Polynomials

If $\alpha+1=-N$, then the weights will be positive iff one of the following eight sets of inequalities holds:

 18.25.1 $\displaystyle-\delta-1$ $\displaystyle<\beta<\gamma+1<-N+1.$ $\displaystyle N-1$ $\displaystyle<-\delta-1<\beta<\gamma+1.$ $\displaystyle\gamma,\delta$ $\displaystyle>-1,\quad\beta>N+\gamma.$ $\displaystyle\gamma,\delta$ $\displaystyle>-1,\quad\beta<-N-\delta.$ $\displaystyle N-1$ $\displaystyle $\displaystyle N+\gamma$ $\displaystyle<\beta<-N-\delta<-N-1.$ $\displaystyle\gamma,\delta$ $\displaystyle<-N,\quad\beta>-1-\delta.$ $\displaystyle\gamma,\delta$ $\displaystyle<-N,\quad\beta<\gamma+1.$ ⓘ Symbols: $N$: positive integer and $\delta$: arbitary small positive constant Referenced by: §18.25(i), Table 18.25.1 Permalink: http://dlmf.nist.gov/18.25.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png See also: Annotations for 18.25(i), 18.25(i), 18.25 and 18

The first four sets imply $\gamma+\delta>-2$, and the last four imply $\gamma+\delta<-2N$.

## §18.25(ii) Weights and Normalizations: Continuous Cases

 18.25.2 $\int_{0}^{\infty}p_{n}(x)p_{m}(x)w(x)\mathrm{d}x=h_{n}\delta_{n,m}.$

### Wilson

 18.25.3 $p_{n}(x)=W_{n}\left(x;a_{1},a_{2},a_{3},a_{4}\right),$
 18.25.4 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $y$: real variable and $w(x)$: weight function Permalink: http://dlmf.nist.gov/18.25.E4 Encodings: TeX, pMML, png See also: Annotations for 18.25(ii), 18.25(ii), 18.25 and 18
 18.25.5 $h_{n}=\frac{n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right)}{(2n-1+% \sum_{j}a_{j})\Gamma\left(n-1+\sum_{j}a_{j}\right)}.$

### Continuous Dual Hahn

 18.25.6 $\displaystyle p_{n}(x)$ $\displaystyle=S_{n}\left(x;a_{1},a_{2},a_{3}\right),$ 18.25.7 $\displaystyle w(y^{2})$ $\displaystyle=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{% \Gamma\left(2iy\right)}\right|^{2},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $y$: real variable and $w(x)$: weight function Permalink: http://dlmf.nist.gov/18.25.E7 Encodings: TeX, pMML, png See also: Annotations for 18.25(ii), 18.25(ii), 18.25 and 18 18.25.8 $\displaystyle h_{n}$ $\displaystyle=n!\,2\pi\prod_{j<\ell}\Gamma\left(n+a_{j}+a_{\ell}\right).$

## §18.25(iii) Weights and Normalizations: Discrete Cases

 18.25.9 $\sum_{y=0}^{N}p_{n}(y(y+\gamma+\delta+1))p_{m}(y(y+\gamma+\delta+1))\*\frac{% \gamma+\delta+1+2y}{\gamma+\delta+1+y}\omega_{y}=h_{n}\delta_{n,m}.$

### Racah

 18.25.10 $p_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta\right),$ $\alpha+1=-N$,
 18.25.11 $\displaystyle\omega_{y}$ $\displaystyle=\frac{{\left(\alpha+1\right)_{y}}{\left(\beta+\delta+1\right)_{y% }}{\left(\gamma+1\right)_{y}}{\left(\gamma+\delta+2\right)_{y}}}{{\left(-% \alpha+\gamma+\delta+1\right)_{y}}{\left(-\beta+\gamma+1\right)_{y}}{\left(% \delta+1\right)_{y}}y!},$ 18.25.12 $\displaystyle h_{n}$ $\displaystyle=\frac{{\left(-\beta\right)_{N}}{\left(\gamma+\delta+2\right)_{N}% }}{{\left(-\beta+\gamma+1\right)_{N}}{\left(\delta+1\right)_{N}}}\frac{{\left(% n+\alpha+\beta+1\right)_{n}}n!}{{\left(\alpha+\beta+2\right)_{2n}}}\*\frac{{% \left(\alpha+\beta-\gamma+1\right)_{n}}{\left(\alpha-\delta+1\right)_{n}}{% \left(\beta+1\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(\beta+\delta+1% \right)_{n}}{\left(\gamma+1\right)_{n}}}.$

### Dual Hahn

 18.25.13 $p_{n}(x)=R_{n}\left(x;\gamma,\delta,N\right),$
 18.25.14 $\omega_{y}=\frac{(-1)^{y}{\left(-N\right)_{y}}{\left(\gamma+1\right)_{y}}{% \left(\gamma+\delta+1\right)_{2}}}{{\left(N+\gamma+\delta+2\right)_{y}}{\left(% \delta+1\right)_{y}}y!},$
 18.25.15 $h_{n}=\frac{n!\,(N-n)!\,{\left(\gamma+\delta+2\right)_{N}}}{N!\,{\left(\gamma+% 1\right)_{n}}{\left(\delta+1\right)_{N-n}}}.$

Table 18.25.2 provides the leading coefficients $k_{n}$18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials.