DLMF:17.2.E42 (Q5332)

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Language	Label	Description	Also known as
English	DLMF:17.2.E42	No description defined

Statements

Defining formula

{\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}}}

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DLMF:17.2.E42

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Defining formula

{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}

C1.S2.SS1.m1amdec

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Defining formula

{\displaystyle{\displaystyle!}}

introduction.Sx4.p1.t1.r15.m5aadec

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Defining formula

{\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}

C17.S2.SS1.p1.m2aaidec

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$$q$$ -binomial coefficient (or Gaussian polynomial)

Defining formula

{\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar% {q}}}}

C17.S2.E27.m2aldec

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Defining formula

{\displaystyle{\displaystyle q}}

C17.S1.XMD10.m1andec

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Defining formula

{\displaystyle{\displaystyle j}}

C17.S1.XMD2.m1idec

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Defining formula

{\displaystyle{\displaystyle n}}

C17.S1.XMD4.m1akdec

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Defining formula

{\displaystyle{\displaystyle z}}

C17.S1.XMD7.m1edec

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$$q$$ -differential operator (locally)

Defining formula

{\displaystyle{\displaystyle\mathcal{D}_{q}}}

C17.S2.XMD1.m1adec

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