Sum[(1/n!) StirlingS1[n, m] z^n w^m, {n, 0, Infinity}, {m, 0, Infinity}] == (z + 1)^w

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], RowBox[List["StirlingS1", "[", RowBox[List["n", ",", "m"]], "]"]], SuperscriptBox["z", "n"], " ", SuperscriptBox["w", "m"]]]]]]], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "w"]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> m </mi> </msup> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> w </mi> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> StirlingS1 </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <ci> w </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> w </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["n_", ",", "m_"]], "]"]], " ", SuperscriptBox["z_", "n_"], " ", SuperscriptBox["w_", "m_"]]], RowBox[List["n_", "!"]]]]]]], "]"]], "\[RuleDelayed]", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "w"]]]]]

2001-10-29