DLMF:17.3.E9 (Q5349)

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DLMF:17.3.E9
No description defined

    Statements

    Defining formula
    a m , s ⁡ ( q ) = q - ( s 2 ) ⁢ ( 1 - q ) s ( q ; q ) s ⁢ ∑ j = 0 s ( - 1 ) j ⁢ q ( j 2 ) ⁢ [ s j ] q ⁢ ( 1 - q s - j ) m ( 1 - q ) m . q-Stirling-number 𝑚 𝑠 𝑞 superscript 𝑞 binomial 𝑠 2 superscript 1 𝑞 𝑠 q-Pochhammer-symbol 𝑞 𝑞 𝑠 superscript subscript 𝑗 0 𝑠 superscript 1 𝑗 superscript 𝑞 binomial 𝑗 2 q-binomial 𝑠 𝑗 𝑞 superscript 1 superscript 𝑞 𝑠 𝑗 𝑚 superscript 1 𝑞 𝑚 {\displaystyle{\displaystyle a_{m,s}\left(q\right)=\frac{q^{-\genfrac{(}{)}{0.% 0pt}{}{s}{2}}(1-q)^{s}}{\left(q;q\right)_{s}}\sum_{j=0}^{s}(-1)^{j}q^{\genfrac% {(}{)}{0.0pt}{}{j}{2}}\genfrac{[}{]}{0.0pt}{}{s}{j}_{q}\frac{(1-q^{s-j})^{m}}{% (1-q)^{m}}.}}
    0 references
    DLMF id
    DLMF:17.3.E9
    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.m1acdec
    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.m2afdec
    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.m2aadec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1hdec
    0 references
    Q11712
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C17.S1.XMD2.m1adec
    0 references
    Q11716
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C17.S1.XMD3.m1adec
    0 references
    Q11717
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C17.S1.XMD6.m1adec
    0 references
     
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