DLMF:17.2.E35 (Q5325)

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DLMF:17.2.E35
No description defined

    Statements

    Defining formula
    ∑ j = 0 n [ n j ] q ⁢ ( - z ) j ⁢ q ( j 2 ) = ( z ; q ) n = ( 1 - z ) ⁢ ( 1 - z ⁢ q ) ⁢ ⋯ ⁢ ( 1 - z ⁢ q n - 1 ) . superscript subscript 𝑗 0 𝑛 q-binomial 𝑛 𝑗 𝑞 superscript 𝑧 𝑗 superscript 𝑞 binomial 𝑗 2 q-Pochhammer-symbol 𝑧 𝑞 𝑛 1 𝑧 1 𝑧 𝑞 ⋯ 1 𝑧 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle\sum_{j=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{j}_{q}(-z% )^{j}q^{\genfrac{(}{)}{0.0pt}{}{j}{2}}=\left(z;q\right)_{n}={(1-z)(1-zq)\cdots% (1-zq^{n-1})}.}}
    0 references
    DLMF id
    DLMF:17.2.E35
    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.m1akdec
    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.m2aaddec
    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.m2ahdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1ahdec
    0 references
    Q11712
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C17.S1.XMD2.m1edec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1aedec
    0 references
    Q11718
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C17.S1.XMD7.m1dec
    0 references
     
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