Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ab8c)

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=2

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

http://functions.wolfram.com/07.27.03.ab8c.01

Input Form

HypergeometricPFQ[{-(5/2), 1, 2}, {1/2, 1/2}, z] == (-(1/512)) I (512 I + 360 Pi^2 Sqrt[z] - 5800 I z - 1800 Pi^2 z^(3/2) + 6300 I z^2 + 1575 Pi^2 z^(5/2)) + (75/128) Sqrt[1 - z] (-10 Sqrt[z] + 21 z^(3/2)) ArcSin[Sqrt[z]] - (1/(1536 (-1 + z)^5)) (Sqrt[1 - z] (-1536 + 48432 z - 282056 z^2 + 747910 z^3 - 1074285 z^4 + 869410 z^5 - 374415 z^6 + 66960 z^7) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(1536 (-1 + z)^5)) (Sqrt[1 - z] (-1536 + 48432 z - 282056 z^2 + 747910 z^3 - 1074285 z^4 + 869410 z^5 - 374415 z^6 + 66960 z^7) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) - (45/128) (8 Sqrt[z] - 40 z^(3/2) + 35 z^(5/2)) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))] + (1/(1536 (-1 + z)^5)) (Sqrt[1 - z] (-1536 + 48432 z - 282056 z^2 + 747910 z^3 - 1074285 z^4 + 869410 z^5 - 374415 z^6 + 66960 z^7) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (45/128) I (8 Sqrt[z] - 40 z^(3/2) + 35 z^(5/2)) PolyLog[2, -E^(I ArcSin[Sqrt[z]])] + (45/128) I (8 Sqrt[z] - 40 z^(3/2) + 35 z^(5/2)) PolyLog[2, E^(I ArcSin[Sqrt[z]])]

Standard Form

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MathML Form

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</msub> <mo> ( </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 512 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1575 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </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["45", "128"], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SqrtBox["z"]]], "-", RowBox[List["40", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["35", " ", 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]