Formula:KLS:14.23:18

<< Formula:KLS:14.23:17

formula in q-Charlier

Formula:KLS:14.23:19 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{0}{1}@@{-}{-a^{-1}q}{q}{-a^{-1}q^{-x+1}t}=\sum_{n=0}^{\infty}\frac{C_{n}\!\left(q^{-x};a;q\right)}{\left(-a^{-1}q,q;q\right)_{n}}t^{n}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle{{}_{r}\phi_{s}}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle C_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Charlier polynomial : http://drmf.wmflabs.org/wiki/Definition:qCharlier

Bibliography

Equation in Section 14.23 of KLS.

URL links

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<< Formula:KLS:14.23:17

formula in q-Charlier

Formula:KLS:14.23:19 >>