# Racah polynomials

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##### 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 $R_{n}\left(x;\gamma,\delta,N\right)$.
###### Further Constraints for RacahPolynomials
18.25.10 $p_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta\right),$ $\alpha+1=-N$,
##### 2: 18.26 Wilson Class: Continued
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 $\gamma+1=-N$; $n=0,1,\dots,N$.
18.26.9 $\lim_{\beta\to\infty}R_{n}\left(x;-N-1,\beta,\gamma,\delta\right)=R_{n}\left(x% ;\gamma,\delta,N\right).$
18.26.10 $\lim_{\delta\to\infty}R_{n}\left(x(x+\gamma+\delta+1);\alpha,\beta,-N-1,\delta% \right)=Q_{n}\left(x;\alpha,\beta,N\right).$
18.26.16 $\frac{\Delta_{y}\left(R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,% \delta\right)\right)}{\Delta_{y}\left(y(y+\gamma+\delta+1)\right)}=\frac{n(n+% \alpha+\beta+1)}{(\alpha+1)(\beta+\delta+1)(\gamma+1)}\*R_{n-1}\left(y(y+% \gamma+\delta+2);\alpha+1,\beta+1,\gamma+1,\delta\right).$
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. …
##### 3: 18.38 Mathematical Applications
###### Coding Theory
For applications of Krawtchouk polynomials $K_{n}\left(x;p,N\right)$ and $q$-Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).
##### 4: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
• Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$.

• $q$-Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$.

• ##### 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 $q$-Racah polynomials form a system of OP’s $\{p_{n}(x)\}$, $n=0,1,2,\dots,N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+cq^{y+1}\}$, $y=0,1,\dots,N$, with $c$ a constant. Both the Askey–Wilson polynomials and the $q$-Racah polynomials can best be described as functions of $z$ (resp. …
###### §18.28(viii) $q$-RacahPolynomials
18.28.19 $R_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)=\sum_{\ell=0}^{n% }\frac{q^{\ell}\left(q^{-n},\alpha\beta q^{n+1};q\right)_{\ell}}{\left(\alpha q% ,\beta\delta q,\gamma q,q;q\right)_{\ell}}\*\prod_{j=0}^{\ell-1}(1-q^{j}x+% \gamma\delta q^{2j+1})={{}_{4}\phi_{3}}\left({q^{-n},\alpha\beta q^{n+1},q^{-y% },\gamma\delta q^{y+1}\atop\alpha q,\beta\delta q,\gamma q};q,q\right),$ $\alpha q$, $\beta\delta q$, or $\gamma q=q^{-N}$; $n=0,1,\dots,N$.
##### 6: 16.4 Argument Unity
One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … …
##### 7: 18.21 Hahn Class: Interrelations Figure 18.21.1: Askey scheme. …It increases by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah polynomials. … Magnify
##### 8: Bibliography C
• L. Chen, M. E. H. Ismail, and P. Simeonov (1999) Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.