Sum[(NorlundB[k, \[Alpha], z] w^k)/k!, {k, 0, Infinity}] == (w/(E^w - 1))^\[Alpha] E^(z w) /; Abs[w] < 2 Pi

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["NorlundB", "[", RowBox[List["k", ",", "\[Alpha]", ",", "z"]], "]"]], " ", SuperscriptBox["w", "k"]]], RowBox[List["k", "!"]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["w", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], "-", "1"]]], ")"]], "\[Alpha]"], SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", "w"]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", RowBox[List["2", "\[Pi]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mrow> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, NorlundB] </annotation> </semantics> <mi> k </mi> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> k </mi> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#63449; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> w </mi> <mrow> <msup> <mi> &#8519; </mi> <mi> w </mi> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> w </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> NorlundB </ci> <ci> k </ci> <ci> &#945; </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#945; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["NorlundB", "[", RowBox[List["k_", ",", "\[Alpha]_", ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "k_"]]], RowBox[List["k_", "!"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["w", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], "-", "1"]]], ")"]], "\[Alpha]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", "w"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", RowBox[List["2", " ", "\[Pi]"]]]]]]]]]]

2007-05-02