Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.ac8g)

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=2, a₃>=2

For fixed z and a₁=-5/2, a₂=2, a₃=2

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

http://functions.wolfram.com/07.27.03.ac8g.01

Input Form

HypergeometricPFQ[{-(5/2), 2, 2}, {-(1/2), 7/2}, z] == (1/(8192 z^2)) (I (17280 I - 1920 I z - 5296 I z^2 - 35200 I z^3 - 11250 Pi^2 z^(7/2) + 44100 I z^4 + 11025 Pi^2 z^(9/2))) - (15 Sqrt[1 - z] (-288 - 64 z - 108 z^2 - 260 z^3 + 735 z^4) ArcSin[Sqrt[z]])/(2048 z^(5/2)) + (1/(16384 (-1 + z)^8)) (Sqrt[1 - z] (128768 - 6709120 z - 37675440 z^2 - 60698720 z^3 + 103843360 z^4 - 179118153 z^5 + 195678150 z^6 - 139851150 z^7 + 63422100 z^8 - 16615425 z^9 + 1920030 z^10) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(16384 (-1 + z)^8)) (Sqrt[1 - z] (-128768 + 6709120 z + 37675440 z^2 + 60698720 z^3 - 103843360 z^4 + 179118153 z^5 - 195678150 z^6 + 139851150 z^7 - 63422100 z^8 + 16615425 z^9 - 1920030 z^10) Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))]) + (225 z^(3/2) (-50 + 49 z) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))])/2048 - (1/(16384 (-1 + z)^8)) (Sqrt[1 - z] (128768 - 6709120 z - 37675440 z^2 - 60698720 z^3 + 103843360 z^4 - 179118153 z^5 + 195678150 z^6 - 139851150 z^7 + 63422100 z^8 - 16615425 z^9 + 1920030 z^10) Log[1 + E^(I ArcSin[Sqrt[z]])]) + (225 I z^(3/2) (-50 + 49 z) PolyLog[2, -E^(I ArcSin[Sqrt[z]])])/2048 - (225 I z^(3/2) (-50 + 49 z) PolyLog[2, E^(I ArcSin[Sqrt[z]])])/2048

Standard Form

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MathML Form

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<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> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 11025 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 9 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 44100 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 11250 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 7 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 35200 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5296 </mn> <mo> ⁢ </mo> <mi> ⅈ 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type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 37675440 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6709120 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 128768 </cn> </apply> <apply> <ln /> <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> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16384 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> 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Rule Form

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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]