Generalized hypergeometric function: Specific values (formula 07.31.03.0140)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Specific values

Specialized values

Case ₇F₆

http://functions.wolfram.com/07.31.03.0140.01

Input Form

HypergeometricPFQ[{1, b, b, b, 1 - b, 1 - b, 1 - b}, {1 + b, 1 + b, 1 + b, 2 - b, 2 - b, 2 - b}, 1] == (((-1 + b)^3 b^3 Pi Csc[b Pi]^3)/(2 (-1 + 2 b)^5)) ((3 + 2 (1 - 2 b)^2 Pi^2) Cos[b Pi] - 3 (Cos[3 b Pi] + 2 (1 - 2 b) Pi Sin[b Pi])) /; b != 1/2

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "b", ",", "b", ",", "b", ",", RowBox[List["1", "-", "b"]], ",", RowBox[List["1", "-", "b"]], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "b"]], ",", RowBox[List["1", "+", "b"]], ",", RowBox[List["1", "+", "b"]], ",", RowBox[List["2", "-", "b"]], ",", RowBox[List["2", "-", "b"]], ",", RowBox[List["2", "-", "b"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], "3"], " ", SuperscriptBox["b", "3"], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], "3"]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], "5"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List["b", "\[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> 7 </mn> </msub> <msub> <mi> F </mi> <mn> 6 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </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["7", TraditionalForm]], SubscriptBox["F", FormBox["6", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "b"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> csc </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> b </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> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <ci> b </ci> <ci> b </ci> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <pi /> <apply> <power /> <apply> <csc /> <apply> <times /> <ci> b </ci> <pi /> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <cos /> <apply> <times /> <ci> b </ci> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> <pi /> </apply> </apply> <apply> <times /> <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> b </ci> </apply> </apply> </apply> <pi /> <apply> <sin /> <apply> <times /> <ci> b </ci> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <neq /> <ci> b </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["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "b_", ",", "b_", ",", "b_", ",", RowBox[List["1", "-", "b_"]], ",", RowBox[List["1", "-", "b_"]], ",", RowBox[List["1", "-", "b_"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "b_"]], ",", RowBox[List["1", "+", "b_"]], ",", RowBox[List["1", "+", "b_"]], ",", RowBox[List["2", "-", "b_"]], ",", RowBox[List["2", "-", "b_"]], ",", RowBox[List["2", "-", "b_"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], "3"], " ", SuperscriptBox["b", "3"], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], "3"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["3", " ", "b", " ", "\[Pi]"]], "]"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], "5"]]]], "/;", RowBox[List["b", "\[NotEqual]", FractionBox["1", "2"]]]]]]]]]

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

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]