Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.als2)

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=3, a₃>=3

For fixed z and a₁=-1/2, a₂=3, a₃=3

For fixed z and a₁=-1/2, a₂=3, a₃=3, b₁=5/2

http://functions.wolfram.com/07.27.03.als2.01

Input Form

HypergeometricPFQ[{-(1/2), 3, 3}, {5/2, 5/2}, z] == -((9 I (-8 Pi^2 - 88 I Sqrt[z] - 72 Pi^2 z + 900 I z^(3/2) + 225 Pi^2 z^2))/ (16384 z^(3/2))) + (9 Sqrt[1 - z] (-14 - 147 z + 225 z^2) ArcSin[Sqrt[z]])/(4096 (-1 + z) z^(3/2)) - (1/(12288 (-1 + z)^6 z)) (Sqrt[1 - z] (-216 + 22248 z - 161357 z^2 + 418808 z^3 - 579087 z^4 + 450114 z^5 - 186420 z^6 + 32130 z^7) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(12288 (-1 + z)^6 z)) (Sqrt[1 - z] (-216 + 22248 z - 161357 z^2 + 418808 z^3 - 579087 z^4 + 450114 z^5 - 186420 z^6 + 32130 z^7) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) - (9 (-8 - 72 z + 225 z^2) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))])/ (4096 z^(3/2)) + (1/(12288 (-1 + z)^6 z)) (Sqrt[1 - z] (-216 + 22248 z - 161357 z^2 + 418808 z^3 - 579087 z^4 + 450114 z^5 - 186420 z^6 + 32130 z^7) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (9 I (-8 - 72 z + 225 z^2) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/ (4096 z^(3/2)) + (9 I (-8 - 72 z + 225 z^2) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/(4096 z^(3/2))

Standard Form

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MathML Form

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</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> <times /> <cn type='integer'> 225 </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'> 147 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -14 </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'> 4096 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> 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</apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 32130 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 186420 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 450114 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 579087 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 418808 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 161357 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> 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RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], RowBox[List["4096", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List["9", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "8"]], "-", RowBox[List["72", " ", "z"]], "+", RowBox[List["225", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]], "]"]]]], RowBox[List["4096", " ", SuperscriptBox["z", RowBox[List["3", "/", "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]