Formula:KLS:14.02:05

<< Formula:KLS:14.02:04

formula in q-Racah

Formula:KLS:14.02:06 >>

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

Substitution(s)

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

${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{\left(\alpha^{-1}\beta^{-1}\gamma,\alpha^{-1}\delta,\beta^{-1},\gamma\delta q^{2};q\right)_{\infty}}{\left(\alpha^{-1}\beta^{-1}q^{-1},\alpha^{-1}\gamma\delta q,\beta^{-1}\gamma q,\delta q;q\right)_{\infty}}{}\frac{(1-\alpha\beta q)(\gamma\delta q)^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta\gamma^{-1}q,\alpha\delta^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\beta\delta q,\gamma q;q\right)_{n}}=\left\{\begin{array}[]{ll}\displaystyle\frac{\left(\beta^{-1},\gamma\delta q^{2};q\right)_{N}}{\left(\beta^{-1}\gamma q,\delta q;q\right)_{N}}\frac{(1-\beta q^{-N})(\gamma\delta q)^{n}}{(1-\beta q^{2n-N})}\frac{\left(q,\beta q,\beta\gamma^{-1}q^{-N},\delta^{-1}q^{-N};q\right)_{n}}{\left(\beta q^{-N},\beta\delta q,\gamma q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\alpha q=q^{-N}\\ \\ \displaystyle\frac{\left(\alpha\beta q^{2},\beta\gamma^{-1};q\right)_{N}}{\left(\alpha\beta\gamma^{-1}q,\beta q;q\right)_{N}}\frac{(1-\alpha\beta q)(\beta^{-1}\gamma q^{-N})^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta q^{N+2},\alpha\beta\gamma^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\gamma q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\beta\delta q=q^{-N}\\ \\ \displaystyle\frac{\left(\alpha\beta q^{2},\delta^{-1};q\right)_{N}}{\left(\alpha\delta^{-1}q,\beta q;q\right)_{N}}\frac{(1-\alpha\beta q)(\delta q^{-N})^{n}}{(1-\alpha\beta q^{2n+1})}\frac{\left(q,\alpha\beta q^{N+2},\alpha\delta^{-1}q,\beta q;q\right)_{n}}{\left(\alpha q,\alpha\beta q,\beta\delta q,q^{-N};q\right)_{n}}&<br/>\quad\textrm{if}\quad\gamma q=q^{-N}\end{array}\right.}}}$ } &
${\displaystyle{\displaystyle{\displaystyle\mu(x):=q^{-x}+\gamma\delta q^{x+1}}}}$ &
${\displaystyle{\displaystyle{\displaystyle\mu(x)=q^{-x}+\gamma\delta q^{x+1}=\lambda(x)=q^{-x}+cq^{x-N}=q^{-x}+q^{x+\gamma+\delta+1}=2a\cos\theta}}}$ &

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

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2

Bibliography

Equation in Section 14.2 of KLS.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:KLS:14.02:04

formula in q-Racah

Formula:KLS:14.02:06 >>