Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ajyq)

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=-1/2, a₂>=-1/2

For fixed z and a₁=-1/2, a₂=1, a₃>=1

For fixed z and a₁=-1/2, a₂=1, a₃=1

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

http://functions.wolfram.com/07.27.03.ajyq.01

Input Form

HypergeometricPFQ[{-(1/2), 1, 1}, {7/2, 7/2}, z] == -((25 I (-144 Pi^2 - 1872 I Sqrt[z] - 648 Pi^2 z - 4736 I z^(3/2) - 162 Pi^2 z^2 + 36 I z^(5/2) + 9 Pi^2 z^3))/(9216 z^(5/2))) + (25 Sqrt[1 - z] (36 + 68 z + z^2) ArcSin[Sqrt[z]])/(256 z^(5/2)) - (1/(3456 (-1 + z)^4 z^2)) (25 Sqrt[1 - z] (-216 - 252 z + 1739 z^2 - 6060 z^3 + 1260 z^4 - 752 z^5 + 81 z^6) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(3456 (-1 + z)^4 z^2)) (25 Sqrt[1 - z] (-216 - 252 z + 1739 z^2 - 6060 z^3 + 1260 z^4 - 752 z^5 + 81 z^6) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) - (25 (-16 - 72 z - 18 z^2 + z^3) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))])/ (256 z^(5/2)) + (1/(3456 (-1 + z)^4 z^2)) (25 Sqrt[1 - z] (-216 - 252 z + 1739 z^2 - 6060 z^3 + 1260 z^4 - 752 z^5 + 81 z^6) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (25 I (-16 - 72 z - 18 z^2 + z^3) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/ (256 z^(5/2)) + (25 I (-16 - 72 z - 18 z^2 + z^3) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/(256 z^(5/2))

Standard Form

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MathML Form

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type='integer'> 144 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 9216 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 25 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 68 </cn> <ci> z </ci> </apply> <cn type='integer'> 36 </cn> </apply> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> 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"3"]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], RowBox[List["256", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]]], "+", FractionBox[RowBox[List["25", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], "-", RowBox[List["72", " ", "z"]], "-", RowBox[List["18", " ", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "3"]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]], "]"]]]], RowBox[List["256", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]]]]]]]]]

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]