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

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=2, a₃>=2

For fixed z and a₁=-1/2, a₂=2, a₃=4

For fixed z and a₁=-1/2, a₂=2, a₃=4, b₁=5/2

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

Input Form

HypergeometricPFQ[{-(1/2), 2, 4}, {5/2, 5/2}, z] == -((3 I (-24 Pi^2 - 200 I Sqrt[z] - 120 Pi^2 z + 1260 I z^(3/2) + 315 Pi^2 z^2))/(8192 z^(3/2))) + (3 Sqrt[1 - z] (-26 - 225 z + 315 z^2) ArcSin[Sqrt[z]])/ (2048 (-1 + z) z^(3/2)) - (1/(51200 (-1 + z)^6 z)) (3 Sqrt[1 - z] (-600 + 26824 z - 265157 z^2 + 714472 z^3 - 997399 z^4 + 776420 z^5 - 321100 z^6 + 55200 z^7) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(51200 (-1 + z)^6 z)) (3 Sqrt[1 - z] (-600 + 26824 z - 265157 z^2 + 714472 z^3 - 997399 z^4 + 776420 z^5 - 321100 z^6 + 55200 z^7) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) - (9 (-8 - 40 z + 105 z^2) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))])/ (2048 z^(3/2)) + (1/(51200 (-1 + z)^6 z)) (3 Sqrt[1 - z] (-600 + 26824 z - 265157 z^2 + 714472 z^3 - 997399 z^4 + 776420 z^5 - 321100 z^6 + 55200 z^7) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (9 I (-8 - 40 z + 105 z^2) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/ (2048 z^(3/2)) + (9 I (-8 - 40 z + 105 z^2) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/(2048 z^(3/2))

Standard Form

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MathML Form

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SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], RowBox[List["2048", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List["9", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "8"]], "-", RowBox[List["40", " ", "z"]], "+", RowBox[List["105", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]], "]"]]]], RowBox[List["2048", " ", 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]