DLMF:17.2.E42 (Q5332)

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DLMF:17.2.E42
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    Statements

    Defining formula
    f [ n ] ⁢ ( z ) = 𝒟 q n ⁢ f ⁢ ( z ) = { z - n ⁢ ( 1 - q ) - n ⁢ ∑ j = 0 n q - n ⁢ j + ( j + 1 2 ) ⁢ ( - 1 ) j ⁢ [ n j ] q ⁢ f ⁢ ( z ⁢ q j ) , z ≠ 0 , f ( n ) ⁢ ( 0 ) ⁢ ( q ; q ) n n ! ⁢ ( 1 - q ) n , z = 0 . superscript 𝑓 delimited-[] 𝑛 𝑧 superscript subscript 𝒟 𝑞 𝑛 𝑓 𝑧 cases superscript 𝑧 𝑛 superscript 1 𝑞 𝑛 superscript subscript 𝑗 0 𝑛 superscript 𝑞 𝑛 𝑗 binomial 𝑗 1 2 superscript 1 𝑗 q-binomial 𝑛 𝑗 𝑞 𝑓 𝑧 superscript 𝑞 𝑗 𝑧 0 superscript 𝑓 𝑛 0 q-Pochhammer-symbol 𝑞 𝑞 𝑛 𝑛 superscript 1 𝑞 𝑛 𝑧 0 {\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}}}
    0 references
    DLMF id
    DLMF:17.2.E42
    0 references
    Symbols used
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1amdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    Q11298
    Defining formula
    ( a ; q ) n q-Pochhammer-symbol 𝑎 𝑞 𝑛 {\displaystyle{\displaystyle\left(\NVar{a};\NVar{q}\right)_{\NVar{n}}}}
    xml-id
    C17.S2.SS1.p1.m2aaidec
    0 references
    $$q$$ -binomial coefficient (or Gaussian polynomial)
    Defining formula
    [ n m ] q q-binomial 𝑛 𝑚 𝑞 {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar% {q}}}}
    xml-id
    C17.S2.E27.m2aldec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1andec
    0 references
    Q11712
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C17.S1.XMD2.m1idec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1akdec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1edec
    0 references
    $$q$$ -differential operator (locally)
    Defining formula
    𝒟 q subscript 𝒟 𝑞 {\displaystyle{\displaystyle\mathcal{D}_{q}}}
    xml-id
    C17.S2.XMD1.m1adec
    0 references
     
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