Formula:KLS:01.15:07

<< Formula:KLS:01.15:06

formula in The q-derivative and q-integral

Formula:KLS:01.15:08 >>

${\displaystyle{\displaystyle{\displaystyle\qderiv[n]{q}@{f(x)g(x)}=\sum_{k=0}^{n}\qBinomial{n}{k}{q}\left(\qderiv[n-k]{q}f\right)(q^{k}x)\left(\qderiv[k]{q}g\right)(x)}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}^{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-derivative : http://drmf.wmflabs.org/wiki/Definition:qderiv
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{0}{}{n}{j}_{q}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-binomial coefficient (or Gaussian polynomial) : http://dlmf.nist.gov/17.2#E27 http://dlmf.nist.gov/26.9#SS2.p1

Bibliography

Equation in Section 1.15 of KLS.

URL links

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<< Formula:KLS:01.15:06

formula in The q-derivative and q-integral

Formula:KLS:01.15:08 >>