Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Hypergeometric2F1Regularized






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric2F1Regularized[a,b,c,z] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/07.24.17.0141.01









  


  










Input Form





Hypergeometric2F1Regularized[a, b, c, z] == Subscript[\[ScriptCapitalC], n][a, b, c, z] Hypergeometric2F1Regularized[a, b, c + n, z] - (((a - n - c) (b - n - c) z)/(z - 1)) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] Hypergeometric2F1Regularized[a, b, c + n + 1, z] /; Subscript[\[ScriptCapitalC], 0][a, b, c, z] == 1 && Subscript[\[ScriptCapitalC], 1][a, b, c, z] == (-c + (1 - a - b + 2 c) z)/ (z - 1) && Subscript[\[ScriptCapitalC], n][a, b, c, z] == ((1 - n - c + (2 (n + c) - 1 - a - b) z)/(z - 1)) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] - (((a - c - n + 1) (b - c - n + 1) z)/(z - 1)) Subscript[\[ScriptCapitalC], n - 2][a, b, c, z] && Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["c", "+", "n"]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "n", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "n", "-", "c"]], ")"]], " ", "z"]], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["c", "+", "n", "+", "1"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "a", "-", "b", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List["z", "-", "1"]]]]], " ", "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "n", "-", "c", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "c"]], ")"]]]], "-", "1", "-", "a", "-", "b"]], ")"]], " ", "z"]]]], RowBox[List[" ", RowBox[List["z", "-", "1"]]]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c", "-", "n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "c", "-", "n", "+", "1"]], ")"]], " ", "z", " "]], RowBox[List["z", "-", "1"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "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> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;c&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;n&quot;]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mrow> <mi> c </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;c&quot;, &quot;+&quot;, &quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;z&quot;, Hypergeometric2F1Regularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mi> c </mi> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mrow> <mtext> </mtext> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> n </ci> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <ci> b </ci> <apply> <plus /> <ci> c </ci> <ci> n </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <ci> b </ci> <apply> <plus /> <ci> c </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 0 </cn> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> n </ci> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <ci> c </ci> </apply> </apply> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["c", "+", "n"]], ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "n", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "n", "-", "c"]], ")"]], " ", "z"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["c", "+", "n", "+", "1"]], ",", "z"]], "]"]]]], RowBox[List["z", "-", "1"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["-", "c"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "a", "-", "b", "+", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List["z", "-", "1"]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "n", "-", "c", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "c"]], ")"]]]], "-", "1", "-", "a", "-", "b"]], ")"]], " ", "z"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], RowBox[List["z", "-", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c", "-", "n", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "c", "-", "n", "+", "1"]], ")"]], " ", "z"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], RowBox[List["z", "-", "1"]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02