DLMF:17.2.E44 (Q5334)

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DLMF:17.2.E44
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    Statements

    Defining formula
    𝒟 q n ⁢ ( f ⁢ ( z ) ⁢ g ⁢ ( z ) ) = ∑ j = 0 n [ n j ] q ⁢ f [ n - j ] ⁢ ( z ⁢ q j ) ⁢ g [ j ] ⁢ ( z ) . superscript subscript 𝒟 𝑞 𝑛 𝑓 𝑧 𝑔 𝑧 superscript subscript 𝑗 0 𝑛 q-binomial 𝑛 𝑗 𝑞 superscript 𝑓 delimited-[] 𝑛 𝑗 𝑧 superscript 𝑞 𝑗 superscript 𝑔 delimited-[] 𝑗 𝑧 {\displaystyle{\displaystyle\mathcal{D}_{q}^{n}(f(z)g(z))=\sum_{j=0}^{n}% \genfrac{[}{]}{0.0pt}{}{n}{j}_{q}f^{[n-j]}(zq^{j})g^{[j]}(z).}}
    0 references
    DLMF id
    DLMF:17.2.E44
    0 references
    Symbols used
    $$q$$ -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.m2amdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1apdec
    0 references
    Q11712
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C17.S1.XMD2.m1jdec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1aldec
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    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1gdec
    0 references
    $$q$$ -differential operator (locally)
    Defining formula
    𝒟 q subscript 𝒟 𝑞 {\displaystyle{\displaystyle\mathcal{D}_{q}}}
    xml-id
    C17.S2.XMD1.m1cdec
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