Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ac8o)

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=2, a₃>=2

For fixed z and a₁=-5/2, a₂=2, a₃=2

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

http://functions.wolfram.com/07.27.03.ac8o.01

Input Form

HypergeometricPFQ[{-(5/2), 2, 2}, {1/2, 3/2}, z] == -((I (5664 I + 3240 Pi^2 Sqrt[z] - 44200 I z - 13500 Pi^2 z^(3/2) + 44100 I z^2 + 11025 Pi^2 z^(5/2)))/6144) + (5 Sqrt[1 - z] (8 - 410 z + 735 z^2) ArcSin[Sqrt[z]])/(512 Sqrt[z]) - (1/(3072 (-1 + z)^5)) (Sqrt[1 - z] (-3072 + 68464 z - 368952 z^2 + 937310 z^3 - 1308690 z^4 + 1037355 z^5 - 439590 z^6 + 77595 z^7) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(3072 (-1 + z)^5)) (Sqrt[1 - z] (-3072 + 68464 z - 368952 z^2 + 937310 z^3 - 1308690 z^4 + 1037355 z^5 - 439590 z^6 + 77595 z^7) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) - (15/512) (72 Sqrt[z] - 300 z^(3/2) + 245 z^(5/2)) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))] + (1/(3072 (-1 + z)^5)) (Sqrt[1 - z] (-3072 + 68464 z - 368952 z^2 + 937310 z^3 - 1308690 z^4 + 1037355 z^5 - 439590 z^6 + 77595 z^7) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (15/512) I (72 Sqrt[z] - 300 z^(3/2) + 245 z^(5/2)) PolyLog[2, -E^(I ArcSin[Sqrt[z]])] + (15/512) I (72 Sqrt[z] - 300 z^(3/2) + 245 z^(5/2)) PolyLog[2, E^(I ArcSin[Sqrt[z]])]

Standard Form

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MathML Form

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type='integer'> -1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <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'> 77595 </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'> 439590 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1037355 </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'> 1308690 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn 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RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["15", "512"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["72", " ", SqrtBox["z"]]], "-", RowBox[List["300", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["245", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", 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]