DLMF:26.10.E2 (Q7885): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12171 / rank
 
Normal rank
Property / Symbols used: Q12171 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12171 / qualifier
 
xml-id: C26.S1.XMD5.m1dec
Property / Symbols used
 
Property / Symbols used: Q12172 / rank
 
Normal rank
Property / Symbols used: Q12172 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12172 / qualifier
 
xml-id: C26.S1.XMD6.m1adec

Latest revision as of 12:59, 2 January 2020

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DLMF:26.10.E2
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    Statements

    test
    ∑ n = 0 ∞ p ⁡ ( 𝒟 , n ) ⁢ q n = ∏ j = 1 ∞ ( 1 + q j ) = ∏ j = 1 ∞ 1 1 - q 2 ⁢ j - 1 = 1 + ∑ m = 1 ∞ q m ⁢ ( m + 1 ) / 2 ( 1 - q ) ⁢ ( 1 - q 2 ) ⁢ ⋯ ⁢ ( 1 - q m ) = 1 + ∑ m = 1 ∞ q m ⁢ ( 1 + q ) ⁢ ( 1 + q 2 ) ⁢ ⋯ ⁢ ( 1 + q m - 1 ) , superscript subscript 𝑛 0 restricted-partitions-P 𝒟 𝑛 superscript 𝑞 𝑛 superscript subscript product 𝑗 1 1 superscript 𝑞 𝑗 superscript subscript product 𝑗 1 1 1 superscript 𝑞 2 𝑗 1 1 superscript subscript 𝑚 1 superscript 𝑞 𝑚 𝑚 1 2 1 𝑞 1 superscript 𝑞 2 ⋯ 1 superscript 𝑞 𝑚 1 superscript subscript 𝑚 1 superscript 𝑞 𝑚 1 𝑞 1 superscript 𝑞 2 ⋯ 1 superscript 𝑞 𝑚 1 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}p\left(\mathcal{D},n\right)q^{n% }=\prod_{j=1}^{\infty}(1+q^{j})=\prod_{j=1}^{\infty}\frac{1}{1-q^{2j-1}}=1+% \sum_{m=1}^{\infty}\frac{q^{m(m+1)/2}}{(1-q)(1-q^{2})\cdots(1-q^{m})}=1+\sum_{% m=1}^{\infty}q^{m}(1+q)(1+q^{2})\cdots\*(1+q^{m-1}),}}
    0 references
    DLMF id
    DLMF:26.10.E2
    0 references
    Symbols used
    Q12191
    test
    p ⁡ ( condition , n ) restricted-partitions-P condition 𝑛 {\displaystyle{\displaystyle p\left(\NVar{\mathrm{condition}},\NVar{n}\right)}}
    xml-id
    C26.S10.SS1.m4abdec
    0 references
    Q12176
    test
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1adec
    0 references
    Q12171
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C26.S1.XMD5.m1dec
    0 references
    Q12172
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1adec
    0 references
     
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