Formula:KLS:14.23:03

<< Formula:KLS:14.23:02

formula in q-Charlier

Formula:KLS:14.23:04 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{a^{x}}{\left(q;q\right)_{x}}q^{\binomial{x}{2}}C_{m}\!\left(q^{-x};a;q\right)C_{n}\!\left(q^{-x};a;q\right){}=q^{-n}\left(-a;q\right)_{\infty}\left(-a^{-1}q,q;q\right)_{n}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle a>0}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle C_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Charlier polynomial : http://drmf.wmflabs.org/wiki/Definition:qCharlier
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.23 of KLS.

URL links

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<< Formula:KLS:14.23:02

formula in q-Charlier

Formula:KLS:14.23:04 >>