Formula:KLS:14.21:04

<< Formula:KLS:14.21:03

formula in q-Laguerre

Formula:KLS:14.21:05 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=-\infty}^{\infty}\frac{q^{k\alpha+k}}{\left(-cq^{k};q\right)_{\infty}}L^{(\alpha)}_{m}\!\left(cq^{k};q\right)L^{(\alpha)}_{n}\!\left(cq^{k};q\right){}=\frac{\left(q,-cq^{\alpha+1},-c^{-1}q^{-\alpha};q\right)_{\infty}}{\left(q^{\alpha+1},-c,-c^{-1}q;q\right)_{\infty}}\frac{\left(q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\alpha>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle c>0}}}$

Proof

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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
${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Laguerre polynomial : http://drmf.wmflabs.org/wiki/Definition:qLaguerre
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.21 of KLS.

URL links

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<< Formula:KLS:14.21:03

formula in q-Laguerre

Formula:KLS:14.21:05 >>