DLMF:26.9.E7 (Q7880): Difference between revisions

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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.m1bdec
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.m1ddec
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.m1cdec
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.m1gdec
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.m1cdec

Latest revision as of 13:58, 2 January 2020

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DLMF:26.9.E7
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    Statements

    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
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    [ n m ] q q-binomial 𝑛 𝑚 𝑞 {\displaystyle{\displaystyle\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{m}}_{\NVar% {q}}}}
    C17.S2.E27.m2acdec
    0 references
    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)}}
    C26.S9.SS1.m1aadec
    0 references
    x 𝑥 {\displaystyle{\displaystyle x}}
    C26.S1.XMD1.m1dec
    0 references
    j 𝑗 {\displaystyle{\displaystyle j}}
    C26.S1.XMD3.m1bdec
    0 references
    k 𝑘 {\displaystyle{\displaystyle k}}
    C26.S1.XMD4.m1ddec
    0 references
    m 𝑚 {\displaystyle{\displaystyle m}}
    C26.S1.XMD5.m1cdec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C26.S1.XMD6.m1gdec
    0 references
    q 𝑞 {\displaystyle{\displaystyle q}}
    C26.S9.XMD1.m1cdec
    0 references