DLMF:26.8.E34 (Q7862): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12131 / rank
 
Normal rank
Property / Symbols used: Q12131 / qualifier
 
test:

S ⁡ ( n , k ) Stirling-number-second-kind-S 𝑛 𝑘 {\displaystyle{\displaystyle S\left(\NVar{n},\NVar{k}\right)}}

\StirlingnumberS@{\NVar{n}}{\NVar{k}}
Property / Symbols used: Q12131 / qualifier
 
xml-id: C26.S8.SS1.p3.m2aqdec
Property / Symbols used
 
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
test:

d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}

\deriv{\NVar{f}}{\NVar{x}}
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S4.E4.m2aadec
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.m1idec
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.m1kdec
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.m1abdec
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.m1agdec

Latest revision as of 12:56, 2 January 2020

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DLMF:26.8.E34
No description defined

    Statements

    test
    ∑ j = 0 n j k ⁢ x j = ∑ j = 0 k S ⁡ ( k , j ) ⁢ x j ⁢ d j d x j ⁡ ( 1 - x n + 1 1 - x ) , superscript subscript 𝑗 0 𝑛 superscript 𝑗 𝑘 superscript 𝑥 𝑗 superscript subscript 𝑗 0 𝑘 Stirling-number-second-kind-S 𝑘 𝑗 superscript 𝑥 𝑗 derivative 𝑥 𝑗 1 superscript 𝑥 𝑛 1 1 𝑥 {\displaystyle{\displaystyle\sum_{j=0}^{n}j^{k}x^{j}=\sum_{j=0}^{k}S\left(k,j% \right)x^{j}\frac{{\mathrm{d}}^{j}}{{\mathrm{d}x}^{j}}\left(\frac{1-x^{n+1}}{1% -x}\right),}}
    0 references
    DLMF id
    DLMF:26.8.E34
    0 references
    Symbols used
    Q12131
    test
    S ⁡ ( n , k ) Stirling-number-second-kind-S 𝑛 𝑘 {\displaystyle{\displaystyle S\left(\NVar{n},\NVar{k}\right)}}
    xml-id
    C26.S8.SS1.p3.m2aqdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    test
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aadec
    0 references
    Q12173
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C26.S1.XMD1.m1idec
    0 references
    Q12176
    test
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1kdec
    0 references
    Q12174
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1abdec
    0 references
    Q12172
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1agdec
    0 references
     
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