Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ahc8)

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.ahc8.01

Input Form

HypergeometricPFQ[{-(3/2), 2, 3}, {1/2, 7/2}, z] == (5 I (-576 I - 288 I z - 2920 I z^2 - 1080 Pi^2 z^(5/2) + 6300 I z^3 + 1575 Pi^2 z^(7/2)))/(16384 z^2) - (15 Sqrt[1 - z] (48 + 40 z - 10 z^2 + 525 z^3) ArcSin[Sqrt[z]])/ (4096 z^(5/2)) - (1/(16384 (-1 + z)^7)) (5 Sqrt[1 - z] (-33024 - 255840 z - 151972 z^2 + 30652 z^3 + 41572 z^4 - 113629 z^5 + 103227 z^6 - 44391 z^7 + 7605 z^8) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(16384 (-1 + z)^7)) (5 Sqrt[1 - z] (-33024 - 255840 z - 151972 z^2 + 30652 z^3 + 41572 z^4 - 113629 z^5 + 103227 z^6 - 44391 z^7 + 7605 z^8) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) + (225 Sqrt[z] (-24 + 35 z) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))])/4096 + (1/(16384 (-1 + z)^7)) (5 Sqrt[1 - z] (-33024 - 255840 z - 151972 z^2 + 30652 z^3 + 41572 z^4 - 113629 z^5 + 103227 z^6 - 44391 z^7 + 7605 z^8) Log[1 + E^(I ArcSin[Sqrt[z]])]) + (225 I Sqrt[z] (-24 + 35 z) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/4096 - (225 I Sqrt[z] (-24 + 35 z) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/4096

Standard Form

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MathML Form

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<mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 30652 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 151972 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 255840 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 33024 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <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> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 225 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 35 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 24 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ 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RowBox[List[RowBox[List["-", "24"]], "+", RowBox[List["35", " ", "z"]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]]]], "]"]]]], "4096"], "-", FractionBox[RowBox[List["225", " ", "\[ImaginaryI]", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "24"]], "+", RowBox[List["35", " ", "z"]]]], ")"]], " ", RowBox[List["PolyLog", "[", RowBox[List["2", ",", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSin", "[", SqrtBox["z"], "]"]]]]]]], "]"]]]], "4096"]]]]]]]

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]