2^z == ContinueFraction[{z, Abs[BernoulliB[k]^z]}, {k, 0, Infinity}] /; Re[z] > 0

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["2", "z"], "\[Equal]", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List["z", ",", RowBox[List["Abs", "[", SuperscriptBox[RowBox[List["BernoulliB", "[", "k", "]"]], "z"], "]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "\[Infinity]"]], "}"]]]], "]"]]]], "/;", "\[InvisibleSpace]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mn> 2 </mn> <mi> z </mi> </msup> <mo> &#10869; </mo> <mrow> <munderover> <mi> &#922; </mi> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mo> &#8290; </mo> <mfrac> <mi> z </mi> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> k </mi> <mi> z </mi> </msubsup> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> Underoverscript </ci> <ci> &#922; </ci> <apply> <ci> Set </ci> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <infinity /> </apply> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <abs /> <apply> <power /> <apply> <ci> BernoulliB </ci> <ci> k </ci> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <times /> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["2", "z_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List["z", ",", RowBox[List["Abs", "[", SuperscriptBox[RowBox[List["BernoulliB", "[", "k", "]"]], "z"], "]"]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "0", ",", "\[Infinity]"]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]

2001-10-29