Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.abl1)

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=4

For fixed z and a₁=-5/2, a₂=1, a₃=4, b₁=-3/2

http://functions.wolfram.com/07.27.03.abl1.01

Input Form

HypergeometricPFQ[{-(5/2), 1, 4}, {-(3/2), 3/2}, z] == (1/(3072 (-1 + z)^2)) (2272 + 4776 z + 45892 z^2 - 17325 I Pi^2 z^(5/2) - 123200 z^3 + 34650 I Pi^2 z^(7/2) + 69300 z^4 - 17325 I Pi^2 z^(9/2)) + (5 Sqrt[1 - z] (-40 - 110 z - 495 z^2 + 5313 z^3 - 8085 z^4 + 3465 z^5) ArcSin[Sqrt[z]])/(768 (-1 + z)^3 Sqrt[z]) - (1/(36864 (-1 + z)^8)) (Sqrt[1 - z] (36864 - 32000 z + 728960 z^2 - 11831760 z^3 + 53887280 z^4 - 128222654 z^5 + 182243715 z^6 - 161479200 z^7 + 87903225 z^8 - 27013950 z^9 + 3599340 z^10) Log[1 - E^(I ArcSin[Sqrt[z]])]) + (1/(36864 (-1 + z)^8)) (Sqrt[1 - z] (36864 - 32000 z + 728960 z^2 - 11831760 z^3 + 53887280 z^4 - 128222654 z^5 + 182243715 z^6 - 161479200 z^7 + 87903225 z^8 - 27013950 z^9 + 3599340 z^10) Log[(1 - E^(I ArcSin[Sqrt[z]]))/ (1 + E^(I ArcSin[Sqrt[z]]))]) - (5775/256) z^(5/2) ArcSin[Sqrt[z]] Log[(1 - E^(I ArcSin[Sqrt[z]]))/(1 + E^(I ArcSin[Sqrt[z]]))] + (1/(36864 (-1 + z)^8)) (Sqrt[1 - z] (36864 - 32000 z + 728960 z^2 - 11831760 z^3 + 53887280 z^4 - 128222654 z^5 + 182243715 z^6 - 161479200 z^7 + 87903225 z^8 - 27013950 z^9 + 3599340 z^10) Log[1 + E^(I ArcSin[Sqrt[z]])]) - (5775/256) I z^(5/2) PolyLog[2, -E^(I ArcSin[Sqrt[z]])] + (5775/256) I z^(5/2) PolyLog[2, E^(I ArcSin[Sqrt[z]])]

Standard Form

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MathML Form

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</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> 5 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3465 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8085 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5313 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 495 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 110 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 40 </mn> </mrow> <mo> ) </mo> </mrow> <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> <mrow> <mn> 768 </mn> <mo> ⁢ 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type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3465 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8085 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5313 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 495 </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'> 110 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -40 </cn> </apply> <apply> <arcsin /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </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]