Formula:KLS:14.02:13

<< Formula:KLS:14.02:12

formula in q-Racah

Formula:KLS:14.02:14 >>

${\displaystyle{\displaystyle{\displaystyle R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)=\frac{\left(\alpha\beta q^{n+1};q\right)_{n}}{\left(\alpha q,\beta\delta q,\gamma q;q\right)_{n}}{\widehat{R}}_{n}\!\left(\mu(x)\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Racah polynomial : http://dlmf.nist.gov/18.28#E19
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{\widehat{R}}_{n}}}}$ : monic ${\displaystyle{\displaystyle{\displaystyle q}}}$-Racah polynomial : http://drmf.wmflabs.org/wiki/Definition:monicqRacah
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.2 of KLS.

URL links

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<< Formula:KLS:14.02:12

formula in q-Racah

Formula:KLS:14.02:14 >>