DLMF:26.10.E3 (Q7886): Difference between revisions

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Property / constraint
 

| x | < 1 𝑥 1 {\displaystyle{\displaystyle|x|<1}}

|x|<1
Property / constraint: | x | < 1 𝑥 1 {\displaystyle{\displaystyle|x|<1}} / rank
 
Normal rank
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q12190 / rank
 
Normal rank
Property / Symbols used: Q12190 / qualifier
 
test:

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)}}

\nrestpartitions[\NVar{k}]@{\leq\NVar{m}}{\NVar{n}}
Property / Symbols used: Q12190 / qualifier
 
xml-id: C26.S9.SS1.m1adec
Property / Symbols used
 
Property / Symbols used: Q12173 / rank
 
Normal rank
Property / Symbols used: Q12173 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12173 / qualifier
 
xml-id: C26.S1.XMD1.m1dec
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.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.m1bdec
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 12:59, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:26.10.E3
No description defined

    Statements

    test
    ( 1 - x ) ⁢ ∑ m , n = 0 ∞ p m ⁡ ( ≤ k , 𝒟 , n ) ⁢ x m ⁢ q n = ∑ m = 0 k [ k m ] q ⁢ q m ⁢ ( m + 1 ) / 2 ⁢ x m = ∏ j = 1 k ( 1 + x ⁢ q j ) , 1 𝑥 superscript subscript 𝑚 𝑛 0 restricted-partitions-P-at-most 𝑚 absent 𝑘 𝒟 𝑛 superscript 𝑥 𝑚 superscript 𝑞 𝑛 superscript subscript 𝑚 0 𝑘 q-binomial 𝑘 𝑚 𝑞 superscript 𝑞 𝑚 𝑚 1 2 superscript 𝑥 𝑚 superscript subscript product 𝑗 1 𝑘 1 𝑥 superscript 𝑞 𝑗 {\displaystyle{\displaystyle(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,% \mathcal{D},n\right)x^{m}q^{n}=\sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}% q^{m(m+1)/2}x^{m}=\prod_{j=1}^{k}(1+x\,q^{j}),}}
    0 references
    DLMF id
    DLMF:26.10.E3
    0 references
    constraint
    | x | < 1 𝑥 1 {\displaystyle{\displaystyle|x|<1}}
    0 references
    Symbols used
    $$q$$ -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.m2adec
    0 references
    Q12190
    test
    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.m1adec
    0 references
    Q12173
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C26.S1.XMD1.m1dec
    0 references
    Q12176
    test
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1bdec
    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.m1bdec
    0 references
    Q12189
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C26.S9.XMD1.m1adec
    0 references
     
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