Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ab1a)

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=1

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

http://functions.wolfram.com/07.27.03.ab1a.01

Input Form

HypergeometricPFQ[{-(5/2), 1, 1}, {-(1/2), 7/2}, z] == (I (1920 I - 4480 I z - 304 I z^2 - 1600 I z^3 - 450 Pi^2 z^(7/2) + 900 I z^4 + 225 Pi^2 z^(9/2)))/(2048 z^2) - (15 Sqrt[1 - z] (-32 + 64 z - 12 z^2 - 20 z^3 + 15 z^4) ArcSin[Sqrt[z]])/ (512 z^(5/2)) + (1/(36864 (-1 + z)^6)) (Sqrt[1 - z] (-86032 + 236196 z - 204626 z^2 - 2752693 z^3 + 5731930 z^4 - 6090930 z^5 + 3661890 z^6 - 1185765 z^7 + 160830 z^8) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(36864 (-1 + z)^6)) (Sqrt[1 - z] (86032 - 236196 z + 204626 z^2 + 2752693 z^3 - 5731930 z^4 + 6090930 z^5 - 3661890 z^6 + 1185765 z^7 - 160830 z^8) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) + (225/512) (-2 + z) z^(3/2) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))] - (1/(36864 (-1 + z)^6)) (Sqrt[1 - z] (-86032 + 236196 z - 204626 z^2 - 2752693 z^3 + 5731930 z^4 - 6090930 z^5 + 3661890 z^6 - 1185765 z^7 + 160830 z^8) Log[1 + E^(I ArcSin[Sqrt[z]])]) + (225/512) I (-2 + z) z^(3/2) PolyLog[2, -E^(I ArcSin[Sqrt[z]])] - (225/512) I (-2 + z) z^(3/2) PolyLog[2, E^(I ArcSin[Sqrt[z]])]

Standard Form

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MathML Form

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"1"]], "+", "z"]], ")"]], "6"]]]], "+", RowBox[List[FractionBox["225", "512"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "z"]], ")"]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], "-", RowBox[List[FractionBox["225", "512"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "z"]], ")"]], " ", SuperscriptBox["z", RowBox[List["3", "/", "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]