Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.7263)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]

Specific values

For integer and half-integer parameters and fixed z

For fixed z and a₁=-7/2, a₂>=-7/2

For fixed z and a₁=-7/2, a₂=1, a₃>=1

For fixed z and a₁=-7/2, a₂=1, a₃=3

For fixed z and a₁=-7/2, a₂=1, a₃=3, b₁=7/2

http://functions.wolfram.com/07.27.03.7263.01

Input Form

HypergeometricPFQ[{-(7/2), 1, 3}, {7/2, 4}, -z] == (-37365 - 78915 z + 108094 z^2 + 67578 z^3 + 23415 z^4 + 3465 z^5)/ (219648 z^2) + (5 (3003 + 6006 z + 9009 z^2 + 8580 z^3 + 5005 z^4 + 1638 z^5 + 231 z^6) ArcTan[Sqrt[z]])/(73216 z^(5/2)) - (5 Log[1 + z])/(143 z^3)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], ",", "1", ",", "3"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "2"], ",", "4"]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "37365"]], "-", RowBox[List["78915", " ", "z"]], "+", RowBox[List["108094", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["67578", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["23415", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3465", " ", SuperscriptBox["z", "5"]]]]], RowBox[List["219648", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List["3003", "+", RowBox[List["6006", " ", "z"]], "+", RowBox[List["9009", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8580", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["5005", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1638", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["231", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", SqrtBox["z"], "]"]]]], RowBox[List["73216", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]]], "-", FractionBox[RowBox[List["5", " ", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]], RowBox[List["143", " ", SuperscriptBox["z", "3"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 4 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["7", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["3", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["7", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["4", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", "z"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> <mo>  </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 3465 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 23415 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 67578 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 108094 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 78915 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 37365 </mn> </mrow> <mrow> <mn> 219648 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 231 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1638 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5005 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8580 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9009 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6006 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 3003 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 73216 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 143 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 7 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> <cn type='integer'> 3 </cn> </list> <list> <cn type='rational'> 7 <sep /> 2 </cn> <cn type='integer'> 4 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3465 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 23415 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 67578 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 108094 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 78915 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -37365 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 219648 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 231 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1638 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5005 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8580 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9009 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6006 </cn> <ci> z </ci> </apply> <cn type='integer'> 3003 </cn> </apply> <apply> <arctan /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 73216 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <ln /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 143 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], ",", "1", ",", "3"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["7", "2"], ",", "4"]], "}"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "37365"]], "-", RowBox[List["78915", " ", "z"]], "+", RowBox[List["108094", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["67578", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["23415", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["3465", " ", SuperscriptBox["z", "5"]]]]], RowBox[List["219648", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List["3003", "+", RowBox[List["6006", " ", "z"]], "+", RowBox[List["9009", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["8580", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["5005", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1638", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["231", " ", SuperscriptBox["z", "6"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", SqrtBox["z"], "]"]]]], RowBox[List["73216", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]]], "-", FractionBox[RowBox[List["5", " ", RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]]]], RowBox[List["143", " ", SuperscriptBox["z", "3"]]]]]]]]]]

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

2007-05-02

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