DLMF:26.9.E5 (Q7878): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12188 / rank
 
Normal rank
Property / Symbols used: Q12188 / qualifier
 
test:

p k ⁡ ( n ) partition-function-at-most 𝑘 𝑛 {\displaystyle{\displaystyle p_{\NVar{k}}\left(\NVar{n}\right)}}

\npartitions[\NVar{k}]@{\NVar{n}}
Property / Symbols used: Q12188 / qualifier
 
xml-id: C26.S9.SS1.p1.m4addec
Property / Symbols used
 
Property / Symbols used: $$q$$ -binomial coefficient (or Gaussian polynomial) / rank
 
Normal rank
Property / Symbols used: $$q$$ -binomial coefficient (or Gaussian polynomial) / qualifier
 
test:

[ n m ] q q-binomial 𝑛 𝑚 𝑞 {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar% {q}}}}

\qbinom{\NVar{n}}{\NVar{m}}{\NVar{q}}
Property / Symbols used: $$q$$ -binomial coefficient (or Gaussian polynomial) / qualifier
 
xml-id: C17.S2.E27.m2aadec
Property / Symbols used
 
Property / Symbols used: Q12176 / rank
 
Normal rank
Property / Symbols used: Q12176 / qualifier
 
test:

j 𝑗 {\displaystyle{\displaystyle j}}

j
Property / Symbols used: Q12176 / qualifier
 
xml-id: C26.S1.XMD3.m1adec
Property / Symbols used
 
Property / Symbols used: Q12174 / rank
 
Normal rank
Property / Symbols used: Q12174 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12174 / qualifier
 
xml-id: C26.S1.XMD4.m1bdec
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.m1adec
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.m1edec
Property / Symbols used
 
Property / Symbols used: Q12189 / rank
 
Normal rank
Property / Symbols used: Q12189 / qualifier
 
test:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q12189 / qualifier
 
xml-id: C26.S9.XMD1.m1adec

Latest revision as of 12:58, 2 January 2020

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DLMF:26.9.E5
No description defined

    Statements

    test
    ∑ n = 0 ∞ p k ⁡ ( n ) ⁢ q n = ∏ j = 1 k 1 1 - q j = 1 + ∑ m = 1 ∞ [ k + m - 1 m ] q ⁢ q m , superscript subscript 𝑛 0 partition-function-at-most 𝑘 𝑛 superscript 𝑞 𝑛 superscript subscript product 𝑗 1 𝑘 1 1 superscript 𝑞 𝑗 1 superscript subscript 𝑚 1 q-binomial 𝑘 𝑚 1 𝑚 𝑞 superscript 𝑞 𝑚 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}p_{k}\left(n\right)q^{n}=\prod_% {j=1}^{k}\frac{1}{1-q^{j}}=1+\sum_{m=1}^{\infty}\genfrac{[}{]}{0.0pt}{}{k+m-1}% {m}_{q}q^{m},}}
    0 references
    DLMF id
    DLMF:26.9.E5
    0 references
    Symbols used
    Q12188
    test
    p k ⁡ ( n ) partition-function-at-most 𝑘 𝑛 {\displaystyle{\displaystyle p_{\NVar{k}}\left(\NVar{n}\right)}}
    xml-id
    C26.S9.SS1.p1.m4addec
    0 references
    $$q$$ -binomial coefficient (or Gaussian polynomial)
    test
    [ 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
    Q12176
    test
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1adec
    0 references
    Q12174
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1bdec
    0 references
    Q12171
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C26.S1.XMD5.m1adec
    0 references
    Q12172
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1edec
    0 references
    Q12189
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C26.S9.XMD1.m1adec
    0 references
     
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