Regularized generalized hypergeometric function: Specific values (formula 07.32.03.0152)

HypergeometricPFQRegularized

$Mathematica Notation$

Hypergeometric Functions

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

Specific values

Specialized values

Case ₉F^~₈

http://functions.wolfram.com/07.32.03.0152.01

Input Form

HypergeometricPFQRegularized[{1, 3/2, 3/2, d, d, d, 1 - d, 1 - d, 1 - d}, {1/2, 1/2, 1 + d, 1 + d, 1 + d, 2 - d, 2 - d, 2 - d}, 1] == (Sin[d Pi]^3/((1 - 2 d)^3 Pi^3)) (Pi (1 - 2 d) Csc[Pi d]^2 + Pi^2 (1 - 2 d)^2 Csc[Pi d]^2 Cot[Pi d] - 2 Cot[Pi d]) /; d != 1/2

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"], ",", FractionBox["3", "2"], ",", "d", ",", "d", ",", "d", ",", RowBox[List["1", "-", "d"]], ",", RowBox[List["1", "-", "d"]], ",", RowBox[List["1", "-", "d"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["1", "+", "d"]], ",", RowBox[List["1", "+", "d"]], ",", RowBox[List["1", "+", "d"]], ",", RowBox[List["2", "-", "d"]], ",", RowBox[List["2", "-", "d"]], ",", RowBox[List["2", "-", "d"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["d", " ", "\[Pi]"]], "]"]], "3"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "d"]]]], ")"]], "3"], " ", SuperscriptBox["\[Pi]", "3"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", "d"]]]], ")"]], SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", "d"]]]], ")"]], "2"], SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]], "2"], RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]]]], "-", RowBox[List["2", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["d", "\[NotEqual]", FractionBox["1", "2"]]]]]]]

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> 9 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 8 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> d </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> d </mi> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> d </mi> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["9", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["8", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["d", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["d", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["d", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "d"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "d"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "d"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["d", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["d", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["d", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "d"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "d"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "d"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> sin </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> d </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> d </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> d </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> d </mi> <mo> ≠ </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <ci> d </ci> <ci> d </ci> <ci> d </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> </apply> </list> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> d </ci> <pi /> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <cot /> <apply> <times /> <pi /> <ci> d </ci> </apply> </apply> <apply> <power /> <apply> <csc /> <apply> <times /> <pi /> <ci> d </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> d </ci> </apply> </apply> </apply> <pi /> <apply> <power /> <apply> <csc /> <apply> <times /> <pi /> <ci> d </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cot /> <apply> <times /> <pi /> <ci> d </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <neq /> <ci> d </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"], ",", FractionBox["3", "2"], ",", "d_", ",", "d_", ",", "d_", ",", RowBox[List["1", "-", "d_"]], ",", RowBox[List["1", "-", "d_"]], ",", RowBox[List["1", "-", "d_"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["1", "+", "d_"]], ",", RowBox[List["1", "+", "d_"]], ",", RowBox[List["1", "+", "d_"]], ",", RowBox[List["2", "-", "d_"]], ",", RowBox[List["2", "-", "d_"]], ",", RowBox[List["2", "-", "d_"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["d", " ", "\[Pi]"]], "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "d"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "d"]]]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]], "2"], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "d"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "d"]]]], ")"]], "3"], " ", SuperscriptBox["\[Pi]", "3"]]]], "/;", RowBox[List["d", "\[NotEqual]", FractionBox["1", "2"]]]]]]]]]

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

2001-10-29

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,a₂},{b₁},z]