Stirling number of the second kind: Summation (formula 04.15.23.0014)

StirlingS2

$Mathematica Notation$

Not involving Stirling numbers of the first kind

http://functions.wolfram.com/04.15.23.0014.01

Input Form

Sum[StirlingS2[n, k] z^k D[f[z], {z, k}], {k, 0, n}] == Nest[Function[f, z D[f, z]], f[z], n] /; Element[n, Integers] && n >= 0

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msubsup> <semantics> <mi> 𝒮 </mi> <annotation encoding='Mathematica'> TagBox["\[ScriptCapitalS]", StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> k </mi> </msup> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mo> ∂ </mo> <mrow> <mo> ∂ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo minsize='3' maxsize='3'> ( </mo> <mrow> <mo minsize='3' maxsize='3'> … </mo> <mo> ⁢ </mo> <mstyle minsize='3' maxsize='3'> <mtext> </mtext> </mstyle> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mo> ∂ </mo> <mrow> <mo> ∂ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <munder> <mo> ︸ </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> times </mi> </mrow> </munder> </munder> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo minsize='3' maxsize='3'> ) </mo> </mrow> <mo minsize='3' maxsize='3'> ) </mo> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msubsup> <semantics> <mi> 𝒮 </mi> <annotation encoding='Mathematica'> TagBox["\[ScriptCapitalS]", StirlingS2] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> k </mi> </msup> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mo> ∂ </mo> <mrow> <mo> ∂ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo minsize='3' maxsize='3'> ( </mo> <mrow> <mo minsize='3' maxsize='3'> … </mo> <mo> ⁢ </mo> <mstyle minsize='3' maxsize='3'> <mtext> </mtext> </mstyle> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mo> ∂ </mo> <mrow> <mo> ∂ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <munder> <mo> ︸ </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> times </mi> </mrow> </munder> </munder> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo minsize='3' maxsize='3'> ) </mo> </mrow> <mo minsize='3' maxsize='3'> ) </mo> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29