CatalanNumber[z] \[Proportional] (2^(2 z)/(Sqrt[Pi] z^(3/2))) Sum[((-1)^k Pochhammer[3/2, k] NorlundB[k, -(1/2), 1/2])/(z^k k!), {k, 0, Infinity}] /; Abs[Arg[z + 1/2]] < Pi && (Abs[z] -> Infinity)

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["2", " ", "z"]]], RowBox[List[SqrtBox["\[Pi]"], SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["NorlundB", "[", RowBox[List["k", ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["z", "+", FractionBox["1", "2"]]], "]"]], "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, CatalanNumber] </annotation> </semantics> <mi> z </mi> </msub> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, NorlundB] </annotation> </semantics> <mi> k </mi> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> CatalanNumber </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 3 <sep /> 2 </cn> <ci> k </ci> </apply> <apply> <ci> NorlundB </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <pi /> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CatalanNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["NorlundB", "[", RowBox[List["k", ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["z", "+", FractionBox["1", "2"]]], "]"]], "]"]], "<", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

2007-05-02