Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ab7w)

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.ab7w.01

Input Form

HypergeometricPFQ[{-(5/2), 1, 2}, {-(1/2), 1/2}, z] == (1/128) I (-128 I - 2200 I z - 900 Pi^2 z^(3/2) + 6300 I z^2 + 1575 Pi^2 z^(5/2)) - (15 Sqrt[1 - z] (6 Sqrt[z] - 95 z^(3/2) + 105 z^(5/2)) ArcSin[Sqrt[z]])/(32 (-1 + z)) + (1/(1536 (-1 + z)^6)) (Sqrt[1 - z] (-1536 - 37392 z + 501912 z^2 - 2217314 z^3 + 4882055 z^4 - 6102720 z^5 + 4419885 z^6 - 1735410 z^7 + 286740 z^8) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(1536 (-1 + z)^6)) (Sqrt[1 - z] (1536 + 37392 z - 501912 z^2 + 2217314 z^3 - 4882055 z^4 + 6102720 z^5 - 4419885 z^6 + 1735410 z^7 - 286740 z^8) Log[(1 - E^(I ArcSin[Sqrt[z]]))/ (1 + E^(I ArcSin[Sqrt[z]]))]) + (225/32) (-4 z^(3/2) + 7 z^(5/2)) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/ (1 + E^(I ArcSin[Sqrt[z]]))] - (1/(1536 (-1 + z)^6)) (Sqrt[1 - z] (-1536 - 37392 z + 501912 z^2 - 2217314 z^3 + 4882055 z^4 - 6102720 z^5 + 4419885 z^6 - 1735410 z^7 + 286740 z^8) Log[1 + E^(I ArcSin[Sqrt[z]])]) + (225/32) I (-4 z^(3/2) + 7 z^(5/2)) PolyLog[2, -E^(I ArcSin[Sqrt[z]])] - (225/32) I (-4 z^(3/2) + 7 z^(5/2)) PolyLog[2, E^(I ArcSin[Sqrt[z]])]

Standard Form

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MathML Form

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4882055 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2217314 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 501912 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 37392 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1536 </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> <mrow> <mfrac> <mn> 225 </mn> <mn> 32 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> Li </mi> <annotation-xml encoding='MathML-Content'> <ci> PolyLog </ci> </annotation-xml> </semantics> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <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> <mfrac> <mn> 225 </mn> <mn> 32 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 3 </mn> <mo> / </mo> 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type='integer'> 15 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 95 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> 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</ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1536 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -286740 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1735410 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn 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type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1536 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 286740 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1735410 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4419885 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6102720 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4882055 </cn> <apply> 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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]