Formula:KLS:14.23:13

<< Formula:KLS:14.23:12

formula in q-Charlier

Formula:KLS:14.23:15 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;a;q)C_{n}\!\left(q^{-x};a;q\right)\right]}{\nabla q^{-x}}=\frac{1}{1-q}w(x;aq;q)C_{n+1}\!\left(q^{-x};aq;q\right)}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x+1}{2}}}{\left(q;q\right)_{x}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle C_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Charlier polynomial : http://drmf.wmflabs.org/wiki/Definition:qCharlier
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1

Bibliography

Equation in Section 14.23 of KLS.

URL links

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<< Formula:KLS:14.23:12

formula in q-Charlier

Formula:KLS:14.23:15 >>