Formula:KLS:14.02:28

<< Formula:KLS:14.02:27

formula in q-Racah

Formula:KLS:14.02:29 >>

${\displaystyle{\displaystyle{\displaystyle\qHyperrphis{2}{1}@@{q^{-x},\beta\gamma^{-1}q^{-x}}{\beta\delta q}{q}{\gamma\delta q^{x+1}t}\ \qHyperrphis{2}{1}@@{\alpha q^{x+1},\gamma q^{x+1}}{\alpha\delta^{-1}q}{q}{q^{-x}t}{}=\sum_{n=0}^{N}\frac{\left(\alpha q,\gamma q;q\right)_{n}}{\left(\alpha\delta^{-1}q,q;q\right)_{n}}R_{n}\!\left(\mu(x);\alpha,\beta,\gamma,\delta\,|\,q\right)t^{n}{}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\textrm{if}\quad\alpha q=q^{-N}\quad\textrm{or}\quad\gamma q=q^{-N}}}}$

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
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : basic hypergeometric (or ${\displaystyle{\displaystyle{\displaystyle q}}}$-hypergeometric) function : http://dlmf.nist.gov/17.4#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-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:14.02:27

formula in q-Racah

Formula:KLS:14.02:29 >>