# §18.26 Wilson Class: Continued

## §18.26(i) Representations as Generalized Hypergeometric Functions and Dualities

For the definition of generalized hypergeometric functions see §16.2. Here we use as convention for (16.2.1) with $b_{q}=-N$, $a_{1}=-n$, and $n=0,1,\ldots,N$ that the summation on the right-hand side ends at $k=n$.

 18.26.1 $W_{n}\left(y^{2};a,b,c,d\right)={\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{% \left(a+d\right)_{n}}\*{{}_{4}F_{3}}\left({-n,n+a+b+c+d-1,a+iy,a-iy\atop a+b,a% +c,a+d};1\right).$
 18.26.2 $S_{n}\left(y^{2};a,b,c\right)={\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{{}_% {3}F_{2}}\left({-n,a+iy,a-iy\atop a+b,a+c};1\right).$
 18.26.3 $R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)={{}_{4}F_{3}% }\left({-n,n+\alpha+\beta+1,-y,y+\gamma+\delta+1\atop\alpha+1,\beta+\delta+1,% \gamma+1};1\right),$ $\alpha+1$ or $\beta+\delta+1$ or $\gamma+1=-N$; $n=0,1,\dots,N$.
 18.26.4 $R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)={{}_{3}F_{2}}\left({-n,% -y,y+\gamma+\delta+1\atop\gamma+1,-N};1\right),$ $n=0,1,\dots,N$.

### Dualities

 18.26.4_1 $R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)=Q_{y}\left(n;\gamma,% \delta,N\right),$ ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$: Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$: dual Hahn polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer Referenced by: §18.26(i), Erratum (V1.2.0) Section 18.26 Permalink: http://dlmf.nist.gov/18.26.E4_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

compare (18.21.1).

 18.26.4_2 $R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)=R_{y}\left(n% (n+\alpha+\beta+1);\gamma,\delta,\alpha,\beta\right).$ ⓘ Symbols: $R_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{% \delta}\right)$: Racah polynomial, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer Referenced by: §18.26(i), Erratum (V1.2.0) Section 18.26 Permalink: http://dlmf.nist.gov/18.26.E4_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

## §18.26(ii) Limit Relations

### Wilson $\to$ Continuous Dual Hahn

 18.26.5 $\lim_{d\to\infty}\frac{W_{n}\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_% {n}\left(x;a,b,c\right).$

### Wilson $\to$ Continuous Hahn

 18.26.6 $\lim_{t\to\infty}\frac{W_{n}\left((x+t)^{2};a-it,b-it,\overline{a}+it,% \overline{b}+it\right)}{(-2t)^{n}n!}=p_{n}\left(x;a,b,\overline{a},\overline{b% }\right).$

### Wilson $\to$ Jacobi

 18.26.7 $\lim_{t\to\infty}\frac{W_{n}\left(\tfrac{1}{2}(1-x)t^{2};\tfrac{1}{2}\alpha+% \tfrac{1}{2},\tfrac{1}{2}\alpha+\tfrac{1}{2},\tfrac{1}{2}\beta+\tfrac{1}{2}+it% ,\tfrac{1}{2}\beta+\tfrac{1}{2}-it\right)}{t^{2n}n!}=P^{(\alpha,\beta)}_{n}% \left(x\right).$

### Continuous Dual Hahn $\to$ Meixner–Pollaczek

 18.26.8 $\lim_{t\to\infty}\ifrac{S_{n}\left((x-t)^{2};\lambda+it,\lambda-it,t\cot\phi% \right)}{t^{n}}=n!(\csc\phi)^{n}P^{(\lambda)}_{n}\left(x;\phi\right).$

### Racah $\to$ Dual Hahn

 18.26.9 $\lim_{\beta\to\infty}R_{n}\left(x;-N-1,\beta,\gamma,\delta\right)=R_{n}\left(x% ;\gamma,\delta,N\right).$

### Racah $\to$ Hahn

 18.26.10 $\lim_{\delta\to\infty}R_{n}\left(x(x+\gamma+\delta+1);\alpha,\beta,-N-1,\delta% \right)=Q_{n}\left(x;\alpha,\beta,N\right).$

### Dual Hahn $\to$ Krawtchouk

 18.26.11 $\lim_{t\to\infty}R_{n}\left(x(x+t+1);pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$

### Dual Hahn $\to$ Meixner

With

 18.26.12 $r(x;\beta,c,N)=x(x+\beta+c^{-1}(1-c)N),$ ⓘ Defines: $r(x;\beta,c,N)$ (locally) Symbols: $N$: positive integer and $x$: real variable Referenced by: §18.26(ii) Permalink: http://dlmf.nist.gov/18.26.E12 Encodings: TeX, pMML, png See also: Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18
 18.26.13 $\lim_{N\to\infty}R_{n}\left(r(x;\beta,c,N);\beta-1,c^{-1}(1-c)N,N\right)=M_{n}% \left(x;\beta,c\right).$

## §18.26(iii) Difference Relations

For comments on the use of the forward-difference operator $\Delta_{x}$, the backward-difference operator $\nabla_{x}$, and the central-difference operator $\delta_{x}$, see §18.2(ii).

For each family only the $y$-difference that lowers $n$ is given. See Koekoek et al. (2010, Chapter 9) for further formulas.

 18.26.14 $\ifrac{\delta_{y}\left(W_{n}\left(y^{2};a,b,c,d\right)\right)}{\delta_{y}(y^{2% })}=-n(n+a+b+c+d-1)\*W_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac% {1}{2},d+\tfrac{1}{2}\right).$ ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\right)$: Wilson polynomial, $\delta_{\NVar{x}}$: central difference, $y$: real variable and $n$: nonnegative integer Referenced by: §16.4(iii), §18.26(iii) Permalink: http://dlmf.nist.gov/18.26.E14 Encodings: TeX, pMML, png See also: Annotations for §18.26(iii), §18.26 and Ch.18
 18.26.15 $\ifrac{\delta_{y}\left(S_{n}\left(y^{2};a,b,c\right)\right)}{\delta_{y}(y^{2})% }=-nS_{n-1}\left(y^{2};a+\tfrac{1}{2},b+\tfrac{1}{2},c+\tfrac{1}{2}\right).$
 18.26.16 $\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,% \delta\right)\right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=\frac{n(n+% \alpha+\beta+1)}{(\alpha+1)(\beta+\delta+1)(\gamma+1)}\*R_{n-1}\left(y(y+% \gamma+\delta+2);\alpha+1,\beta+1,\gamma+1,\delta\right).$
 18.26.17 $\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)% \right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=-\frac{n}{(\gamma+1)N}\*% R_{n-1}\left(y(y+\gamma+\delta+2);\gamma+1,\delta,N-1\right).$

## §18.26(iv) Generating Functions

For the hypergeometric function ${{}_{2}F_{1}}$ see §§15.1 and 15.2(i).

### Wilson

 18.26.18 ${{}_{2}F_{1}}\left({a+\mathrm{i}y,d+\mathrm{i}y\atop a+d};z\right){{}_{2}F_{1}% }\left({b-\mathrm{i}y,c-\mathrm{i}y\atop b+c};z\right)=\sum_{n=0}^{\infty}% \frac{W_{n}\left(y^{2};a,b,c,d\right)}{{\left(a+d\right)_{n}}{\left(b+c\right)% _{n}}n!}z^{n},$ $|z|<1$.

### Continuous Dual Hahn

 18.26.19 $(1-z)^{-c+\mathrm{i}y}{{}_{2}F_{1}}\left({a+\mathrm{i}y,b+\mathrm{i}y\atop a+b% };z\right)=\sum_{n=0}^{\infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b% \right)_{n}}n!}z^{n},$ $|z|<1$.

### Racah

 18.26.20 ${{}_{2}F_{1}}\left({-y,-y+\beta-\gamma\atop\beta+\delta+1};z\right){{}_{2}F_{1% }}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}^{N}\frac{{\left(-N% \right)_{n}}{\left(\gamma+1\right)_{n}}}{{\left(-\delta-N\right)_{n}}n!}R_{n}% \left(y(y+\gamma+\delta+1);-N-1,\beta,\gamma,\delta\right)z^{n}.$

### Dual Hahn

 18.26.21 $(1-z)^{y}{{}_{2}F_{1}}\left({y-N,y+\gamma+1\atop-\delta-N};z\right)=\sum_{n=0}% ^{N}\frac{{\left(\gamma+1\right)_{n}}{\left(-N\right)_{n}}}{{\left(-\delta-N% \right)_{n}}n!}\*R_{n}\left(y(y+\gamma+\delta+1);\gamma,\delta,N\right)z^{n}.$

## §18.26(v) Asymptotic Approximations

For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998).

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. Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.