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

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=-3/2, a₂>=-3/2

For fixed z and a₁=-3/2, a₂=2, a₃>=2

For fixed z and a₁=-3/2, a₂=2, a₃=3

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

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

Input Form

HypergeometricPFQ[{-(3/2), 2, 3}, {-(1/2), 7/2}, z] == -((5 I (1728 I + 672 I z + 1400 I z^2 + 6300 I z^3 + 1575 Pi^2 z^(7/2)))/ (8192 z^2)) + (15 Sqrt[1 - z] (144 - 40 z - 70 z^2 - 175 z^3 + 525 z^4) ArcSin[Sqrt[z]])/(2048 (-1 + z) z^(5/2)) + (1/(16384 (-1 + z)^8)) (5 Sqrt[1 - z] (-33024 + 713856 z + 3562536 z^2 + 1231640 z^3 - 252720 z^4 + 401529 z^5 - 369761 z^6 + 208089 z^7 - 65655 z^8 + 8910 z^9) Log[1 - E^(I ArcSin[Sqrt[z]])]) - (1/(16384 (-1 + z)^8)) (5 Sqrt[1 - z] (-33024 + 713856 z + 3562536 z^2 + 1231640 z^3 - 252720 z^4 + 401529 z^5 - 369761 z^6 + 208089 z^7 - 65655 z^8 + 8910 z^9) Log[(1 - E^(I ArcSin[Sqrt[z]]))/ (1 + E^(I ArcSin[Sqrt[z]]))]) - (7875 z^(3/2) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/ (1 + E^(I ArcSin[Sqrt[z]]))])/2048 - (1/(16384 (-1 + z)^8)) (5 Sqrt[1 - z] (-33024 + 713856 z + 3562536 z^2 + 1231640 z^3 - 252720 z^4 + 401529 z^5 - 369761 z^6 + 208089 z^7 - 65655 z^8 + 8910 z^9) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (7875 I z^(3/2) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/2048 + (7875 I z^(3/2) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/2048

Standard Form

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MathML Form

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RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"]]]], "-", FractionBox[RowBox[List["7875", " ", "\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], "2048"], "+", FractionBox[RowBox[List["7875", " ", "\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]], "]"]]]], "2048"]]]]]]]

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]