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 $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. ►

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

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$p_n(x)$	$x=\lambda (y)$	Orthogonality range for $y$	Constraints
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$R_n(x;\alpha ,\beta ,\gamma ,\delta )$	$y(y+\gamma +\delta +1)$	$\{0,1,\mathrm{\dots },N\}$	$\alpha +1$ or $\beta +\delta +1$ or $\gamma +1=-N;$ for further constraints see (18.25.1)
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Further Constraints for Racah Polynomials

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18.25.10 $$p_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta ),$$ $\alpha +1=-N$,

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant, $p_n(x)$: polynomial of degree $n$ and $x$: real variable

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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Table 18.25.2: Wilson class OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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$R_n(x;\alpha ,\beta ,\gamma ,\delta )$	${\displaystyle \frac{{\left(n+\alpha +\beta +1\right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +\delta +1\right)}_n{\left(\gamma +1\right)}_n}}$
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2: 18.26 Wilson Class: Continued

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18.26.3 $$R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )={}_4{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +\beta +1,-y,y+\gamma +\delta +1}{\alpha +1,\beta +\delta +1,\gamma +1};1),$$ $\alpha +1$ or $\beta +\delta +1$ or $\gamma +1=-N$; $n=0,1,\mathrm{\dots },N$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $y$: real variable, $n$: nonnegative integer, $N$: positive integer and $\delta$: arbitary small positive constant

Referenced by:

§18.26(i), §18.26(ii), §18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E3

Encodings:

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See also:

Annotations for §18.26(i), §18.26 and Ch.18

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18.26.9 $$\underset{\beta \to \mathrm{\infty }}{lim}{R}_n(x;-N-1,\beta ,\gamma ,\delta )=R_n(x;\gamma ,\delta ,N).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E9

Encodings:

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See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.10 $$\underset{\delta \to \mathrm{\infty }}{lim}{R}_n(x(x+\gamma +\delta +1);\alpha ,\beta ,-N-1,\delta )=Q_n(x;\alpha ,\beta ,N).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E10

Encodings:

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See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.16 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=\frac{n(n+\alpha +\beta +1)}{(\alpha +1)(\beta +\delta +1)(\gamma +1)}{R}_{n-1}(y(y+\gamma +\delta +2);\alpha +1,\beta +1,\gamma +1,\delta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $n$: nonnegative integer and $\delta$: arbitary small positive constant

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.26.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iii), §18.26 and Ch.18

… ►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

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Coding Theory

►For applications of Krawtchouk polynomials $K_n(x;p,N)$ and $q$-Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).

4: 18.1 Notation

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$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

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Racah: $R_n(x;\alpha ,\beta ,\gamma ,\delta )$.

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$q$-Racah: $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$.

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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 $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, 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) $q$-Racah Polynomials

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18.28.19 $$R_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta |q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{\mathrm{\ell }}{(q^{-n},\alpha \beta q^{n+1};q)}_{\mathrm{\ell }}}{{(\alpha q,\beta \delta q,\gamma q,q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-q^{j}x+\gamma \delta q^{2j+1})={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},\alpha \beta q^{n+1},q^{-y},\gamma \delta q^{y+1}}{\alpha q,\beta \delta q,\gamma q};q,q),$$ $\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n=0,1,\mathrm{\dots },N$.

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Defines:

$R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial and $R_n(x)$ (locally)

Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $N$: positive integer, $\delta$: arbitary small positive constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E19

Encodings:

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See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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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

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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

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Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

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