DLMF:26.8.E13 (Q7838)

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DLMF:26.8.E13
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    Statements

    Defining formula
    ∑ n , k = 0 ∞ S ⁡ ( n , k ) ⁢ x n n ! ⁢ y k = exp ⁡ ( y ⁢ ( e x - 1 ) ) . superscript subscript 𝑛 𝑘 0 Stirling-number-second-kind-S 𝑛 𝑘 superscript 𝑥 𝑛 𝑛 superscript 𝑦 𝑘 𝑦 𝑥 1 {\displaystyle{\displaystyle\sum_{n,k=0}^{\infty}S\left(n,k\right)\frac{x^{n}}% {n!}y^{k}=\exp\left(y({\mathrm{e}^{x}}-1)\right).}}
    0 references
    DLMF id
    DLMF:26.8.E13
    0 references
    Symbols used
    Q12131
    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.m2agdec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
     
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