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DLMF:26.8.E34 - MaRDI portal

DLMF:26.8.E34 (Q8555)

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DLMF:26.8.E34
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    Statements

    DLMF defining formula
    ∑ 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
    Symbols used
    Q12134
    DLMF defining formula
    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 $$x$$
    DLMF defining formula
    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
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C26.S1.XMD1.m1idec
    0 references
    nonnegative integer
    DLMF defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C26.S1.XMD3.m1kdec
    0 references
    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1abdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1agdec
    0 references

    Identifiers

     
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