Formula:6029

Digital Library of Mathematical Functions ID 17.2.E42

${\displaystyle{\displaystyle f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}}}$

Symbols List

• ${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar{q}}}}$: $$q$$ -binomial coefficient (or Gaussian polynomial)
• ${\displaystyle{\displaystyle q}}$: complex base
• ${\displaystyle{\displaystyle j}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\mathcal{D}_{q}}}$: $$q$$ -differential operator (locally)
 

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