Formula:KLS:14.02:22

<< Formula:KLS:14.02:21

formula in q-Racah

Formula:KLS:14.02:23 >>

${\displaystyle{\displaystyle{\displaystyle(1-\alpha q^{x})(1-\beta\delta q^{x})(1-\gamma q^{x})(1-\gamma\delta q^{x})R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right){}-(1-q^{x})(1-\delta q^{x})(\alpha-\gamma\delta q^{x})(\beta-\gamma q^{x})R_{n}\!\left(\mu(x-1);\alpha,\beta,\gamma,\delta\,|\,q\right){}=q^{x}(1-\alpha)(1-\beta\delta)(1-\gamma)(1-\gamma\delta q^{2x}){}R_{n+1}\!\left(\mu(x);\alpha q^{-1},\beta q^{-1},\gamma q^{-1},\delta\,|\,q\right)}}}$

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
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : 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:14.02:21

formula in q-Racah

Formula:KLS:14.02:23 >>