Regularized hypergeometric function 2F1: Identities (formula 07.24.17.0116)

Hypergeometric2F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1Regularized[a,b,c,z]

Identities

Functional identities

Cubic transformations

http://functions.wolfram.com/07.24.17.0116.01

Input Form

Hypergeometric2F1Regularized[a, a + 1/2, (4 a + 5)/6, z] == Hypergeometric2F1Regularized[a/3, a/3 + 1/2, (4 a + 5)/6, -((27 z (1 - z)^2)/(1 - 9 z)^2)]/(1 - 9 z)^(2 (a/3)) /; Abs[z] < 1/9

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", FractionBox["1", "2"]]], ",", FractionBox[RowBox[List[RowBox[List["4", "a"]], "+", "5"]], "6"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["9", "z"]]]], ")"]], RowBox[List[RowBox[List["-", "2"]], RowBox[List["a", "/", "3"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[FractionBox["a", "3"], ",", RowBox[List[FractionBox["a", "3"], "+", FractionBox["1", "2"]]], ",", FractionBox[RowBox[List[RowBox[List["4", "a"]], "+", "5"]], "6"], ",", RowBox[List["-", FractionBox[RowBox[List["27", "z", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["9", "z"]]]], ")"]], "2"]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", FractionBox["1", "9"]]]]]]]

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>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mn> 6 </mn> </mfrac> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", "a"]], "+", "5"]], "6"], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mn> 6 </mn> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 27 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["a", "3"], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["a", "3"], "+", FractionBox["1", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["4", "a"]], "+", "5"]], "6"], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["27", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["9", " ", "z"]]]], ")"]], "2"]]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mfrac> <mn> 1 </mn> <mn> 9 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Hypergeometric2F1Regularized </ci> <ci> a </ci> <apply> <plus /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 27 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 9 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", RowBox[List["a_", "+", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a_"]], "+", "5"]], ")"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["9", " ", "z"]]]], ")"]], RowBox[List["-", FractionBox[RowBox[List["2", " ", "a"]], "3"]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[FractionBox["a", "3"], ",", RowBox[List[FractionBox["a", "3"], "+", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "a"]], "+", "5"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["27", " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["9", " ", "z"]]]], ")"]], "2"]]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", FractionBox["1", "9"]]]]]]]]]

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

2001-10-29

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]