DLMF:26.9.E7 (Q7880)

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DLMF:26.9.E7
No description defined

    Statements

    Defining formula
    ∑ m , n = 0 ∞ p k ⁡ ( ≤ m , n ) ⁢ x k ⁢ q n = 1 + ∑ k = 1 ∞ [ m + k k ] q ⁢ x k = ∏ j = 0 m 1 1 - x ⁢ q j . superscript subscript 𝑚 𝑛 0 restricted-partitions-P-at-most 𝑘 absent 𝑚 𝑛 superscript 𝑥 𝑘 superscript 𝑞 𝑛 1 superscript subscript 𝑘 1 q-binomial 𝑚 𝑘 𝑘 𝑞 superscript 𝑥 𝑘 superscript subscript product 𝑗 0 𝑚 1 1 𝑥 superscript 𝑞 𝑗 {\displaystyle{\displaystyle\sum_{m,n=0}^{\infty}p_{k}\left(\leq m,n\right)x^{% k}q^{n}=1+\sum_{k=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{m+k}{k}_{q}x^{k}=\prod_{j% =0}^{m}\frac{1}{1-x\,q^{j}}.}}
    0 references
    DLMF id
    DLMF:26.9.E7
    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.m2acdec
    0 references
    Q12190
    Defining formula
    p k ⁡ ( ≤ m , n ) restricted-partitions-P-at-most 𝑘 absent 𝑚 𝑛 {\displaystyle{\displaystyle p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)}}
    xml-id
    C26.S9.SS1.m1aadec
    0 references
    Q12173
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C26.S1.XMD1.m1dec
    0 references
    Q12176
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1bdec
    0 references
    Q12174
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1ddec
    0 references
    Q12171
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C26.S1.XMD5.m1cdec
    0 references
    Q12172
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1gdec
    0 references
    Q12189
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C26.S9.XMD1.m1cdec
    0 references
     
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