DLMF:26.8.E12 (Q7837)

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DLMF:26.8.E12
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    Statements

    test
    ∑ n = 0 ∞ S ⁡ ( n , k ) ⁢ x n n ! = ( e x - 1 ) k k ! , superscript subscript 𝑛 0 Stirling-number-second-kind-S 𝑛 𝑘 superscript 𝑥 𝑛 𝑛 superscript 𝑥 1 𝑘 𝑘 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}S\left(n,k\right)\frac{x^{n}}{n% !}=\frac{({\mathrm{e}^{x}}-1)^{k}}{k!},}}
    0 references
    DLMF id
    DLMF:26.8.E12
    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.m2afdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
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    Q12173
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C26.S1.XMD1.m1edec
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    Q12174
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C26.S1.XMD4.m1kdec
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    Q12172
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1ldec
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