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variants of this functions
GegenbauerC




Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > General case





http://functions.wolfram.com/07.14.06.0002.01









  


  








Input Form





GegenbauerC[\[Nu], \[Lambda], z] == ((2^(1 - 2 \[Lambda]) Sqrt[Pi] Gamma[\[Nu] + 2 \[Lambda]])/ (Gamma[\[Nu] + 1] Gamma[\[Lambda]])) Sum[(Pochhammer[-\[Nu], j + k] Pochhammer[2 \[Lambda] + \[Nu], j + k] (-z)^j)/(Gamma[\[Lambda] + 1/2 + j + k] j! k! 2^(j + k)), {j, 0, Infinity}, {k, 0, Infinity}] /; Abs[z] < 1








Standard Form





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MathML Form







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</mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> </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> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> &#957; </ci> </apply> <ci> &#955; </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> &#955; </ci> </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> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> j </ci> <ci> k </ci> <ci> &#955; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "\[Lambda]"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "\[Lambda]"]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["j", "+", "k"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], ",", RowBox[List["j", "+", "k"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "j"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", FractionBox["1", "2"], "+", "j", "+", "k"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["2", RowBox[List["j", "+", "k"]]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]


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





2001-10-29