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

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=-5/2, a₂>=-5/2

For fixed z and a₁=-5/2, a₂=1, a₃>=1

For fixed z and a₁=-5/2, a₂=1, a₃=3

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

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

Input Form

HypergeometricPFQ[{-(5/2), 1, 3}, {3/2, 3/2}, z] == -((1/(8192 Sqrt[z])) (I (-240 Pi^2 + 5072 I Sqrt[z] + 1800 Pi^2 z - 11200 I z^(3/2) - 3150 Pi^2 z^2 + 6300 I z^(5/2) + 1575 Pi^2 z^3))) + (15 Sqrt[1 - z] (36 - 140 z + 105 z^2) ArcSin[Sqrt[z]])/(2048 Sqrt[z]) - (1/(6144 (-1 + z)^4)) (Sqrt[1 - z] (5424 - 44264 z + 136886 z^2 - 216011 z^3 + 186620 z^4 - 84405 z^5 + 15690 z^6) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(6144 (-1 + z)^4)) (Sqrt[1 - z] (5424 - 44264 z + 136886 z^2 - 216011 z^3 + 186620 z^4 - 84405 z^5 + 15690 z^6) Log[(1 - E^(I ArcSin[Sqrt[z]]))/ (1 + E^(I ArcSin[Sqrt[z]]))]) - (15 (-16 + 120 z - 210 z^2 + 105 z^3) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))])/ (2048 Sqrt[z]) + (1/(6144 (-1 + z)^4)) (Sqrt[1 - z] (5424 - 44264 z + 136886 z^2 - 216011 z^3 + 186620 z^4 - 84405 z^5 + 15690 z^6) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (15 I (-16 + 120 z - 210 z^2 + 105 z^3) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/ (2048 Sqrt[z]) + (15 I (-16 + 120 z - 210 z^2 + 105 z^3) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/(2048 Sqrt[z])

Standard Form

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MathML Form

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type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 11200 </cn> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1800 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 5072 </cn> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 240 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8192 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15 </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'> 105 </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'> 140 </cn> <ci> z </ci> </apply> </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'> 2048 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <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> <power /> <apply> 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RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], RowBox[List["2048", " ", SqrtBox["z"]]]], "+", FractionBox[RowBox[List["15", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], "+", RowBox[List["120", " ", "z"]], "-", RowBox[List["210", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["105", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]], "]"]]]], RowBox[List["2048", " ", SqrtBox["z"]]]]]]]]]]

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]