Racah polynomials

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1: 18.26 Wilson Class: Continued
18.26.4_2 $R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,\gamma,\delta\right)=R_{y}\left(n% (n+\alpha+\beta+1);\gamma,\delta,\alpha,\beta\right).$
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. …
2: 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 $p_{n}(x)=R_{n}\left(x;\alpha,\beta,\gamma,\delta\right),$ $\alpha+1=-N$,
Both the Askey–Wilson polynomials and the $q$-Racah polynomials can best be described as functions of $z$ (resp. …
§18.28(viii) $q$-RacahPolynomials
18.28.23 $R_{n}\left(q^{-y}+\gamma\delta q^{y+1};\alpha,\beta,\gamma,\delta\,|\,q\right)% =R_{y}\left(q^{-n}+\alpha\beta q^{n+1};\gamma,\delta,\alpha,\beta\,|\,q\right),$ $\alpha q$, $\beta\delta q$, or $\gamma q=q^{-N}$; $n,y=0,1,\ldots,N$.
These systems are the $q$-Racah polynomials and its limit cases. …
18.28.34 $\lim_{q\to 1}R_{n}\left(q^{-y}+q^{y+\gamma+\delta+1};q^{\alpha},q^{\beta},q^{% \gamma},q^{\delta}\,|\,q\right)=R_{n}\left(y(y+\gamma+\delta+1);\alpha,\beta,% \gamma,\delta\right).$
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: 18.38 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). … The $6j$ symbol (34.4.3), with an alternative expression as a terminating balanced ${{}_{4}F_{3}}$ of unit argument, can be expressend in terms of Racah polynomials (18.26.3). The orthogonality relations (34.5.14) for the $6j$ symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … …
6: 16.4 Argument Unity
One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … …
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.