DLMF:27.14.E4 (Q8103): 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: Euler’s reciprocal function / rank
 
Normal rank
Property / Symbols used: Euler’s reciprocal function / qualifier
 
Defining formula:

f ⁡ ( x ) Euler-phi 𝑥 {\displaystyle{\displaystyle\mathit{f}\left(\NVar{x}\right)}}

\EulerPhi@{\NVar{x}}
Property / Symbols used: Euler’s reciprocal function / qualifier
 
xml-id: C27.S14.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: Q12238 / rank
 
Normal rank
Property / Symbols used: Q12238 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q12238 / qualifier
 
xml-id: C27.S1.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q12237 / rank
 
Normal rank
Property / Symbols used: Q12237 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12237 / qualifier
 
xml-id: C27.S1.XMD6.m1bdec
Property / Symbols used
 
Property / Symbols used: pentagonal numbers (locally) / rank
 
Normal rank
Property / Symbols used: pentagonal numbers (locally) / qualifier
 
Defining formula:

ω ⁢ ( k ) 𝜔 𝑘 {\displaystyle{\displaystyle\omega(k)}}

\omega(k)
Property / Symbols used: pentagonal numbers (locally) / qualifier
 
xml-id: C27.S14.XMD1.m1dec

Latest revision as of 14:30, 2 January 2020

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DLMF:27.14.E4
No description defined

    Statements

    Defining formula
    f ⁡ ( x ) = 1 - x - x 2 + x 5 + x 7 - x 12 - x 15 + … = 1 + ∑ k = 1 ∞ ( - 1 ) k ⁢ ( x ω ⁢ ( k ) + x ω ⁢ ( - k ) ) , Euler-phi 𝑥 1 𝑥 superscript 𝑥 2 superscript 𝑥 5 superscript 𝑥 7 superscript 𝑥 12 superscript 𝑥 15 … 1 superscript subscript 𝑘 1 superscript 1 𝑘 superscript 𝑥 𝜔 𝑘 superscript 𝑥 𝜔 𝑘 {\displaystyle{\displaystyle\mathit{f}\left(x\right)=1-x-x^{2}+x^{5}+x^{7}-x^{% 12}-x^{15}+\dots=1+\sum_{k=1}^{\infty}(-1)^{k}\left(x^{\omega(k)}+x^{\omega(-k% )}\right),}}
    0 references
    DLMF id
    DLMF:27.14.E4
    0 references
    constraint
    | x | < 1 𝑥 1 {\displaystyle{\displaystyle|x|<1}}
    0 references
    Symbols used
    Euler’s reciprocal function
    Defining formula
    f ⁡ ( x ) Euler-phi 𝑥 {\displaystyle{\displaystyle\mathit{f}\left(\NVar{x}\right)}}
    xml-id
    C27.S14.E2.m2abdec
    0 references
    Q12238
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C27.S1.XMD2.m1dec
    0 references
    Q12237
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C27.S1.XMD6.m1bdec
    0 references
    pentagonal numbers (locally)
    Defining formula
    ω ⁢ ( k ) 𝜔 𝑘 {\displaystyle{\displaystyle\omega(k)}}
    xml-id
    C27.S14.XMD1.m1dec
    0 references
     
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