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CatalanNumber






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > CatalanNumber[n] > Series representations > Generalized power series > Expansions at z==z0/;z0!=-n-1/2





http://functions.wolfram.com/06.41.06.0005.01









  


  










Input Form





CatalanNumber[z] \[Proportional] CatalanNumber[Subscript[z, 0]] (1 + (Log[4] + PolyGamma[1/2 + Subscript[z, 0]] - PolyGamma[2 + Subscript[z, 0]]) (z - Subscript[z, 0]) + (1/2) ((Log[4] + PolyGamma[1/2 + Subscript[z, 0]] - PolyGamma[2 + Subscript[z, 0]])^2 + PolyGamma[1, 1/2 + Subscript[z, 0]] - PolyGamma[1, 2 + Subscript[z, 0]]) (z - Subscript[z, 0])^2 + \[Ellipsis]) /; (z -> Subscript[z, 0]) && !(Element[-Subscript[z, 0] - 1/2, Integers] && -Subscript[z, 0] - 1/2 >= 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["CatalanNumber", "[", SubscriptBox["z", "0"], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["z", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", SubscriptBox["z", "0"]]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], " ", "+", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["z", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", SubscriptBox["z", "0"]]], "]"]]]], ")"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["z", "0"]]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["2", "+", SubscriptBox["z", "0"]]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "->", SubscriptBox["z", "0"]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["z", "0"]]], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["z", "0"]]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]










MathML Form







<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> <msub> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, CatalanNumber] </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </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> <ci> CatalanNumber </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["CatalanNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["CatalanNumber", "[", SubscriptBox["zz", "0"], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["zz", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", SubscriptBox["zz", "0"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["zz", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", SubscriptBox["zz", "0"]]], "]"]]]], ")"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[FractionBox["1", "2"], "+", SubscriptBox["zz", "0"]]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["2", "+", SubscriptBox["zz", "0"]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["zz", "0"]]], "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["zz", "0"]]], "-", FractionBox["1", "2"]]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]










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





2007-05-02





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