Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.a9n1)

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=-1/2, a₃>=-1/2

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

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

http://functions.wolfram.com/07.27.03.a9n1.01

Input Form

HypergeometricPFQ[{-(5/2), -(1/2), 1/2}, {4, 4}, z] == -((1/(420217875 Pi^2 z^3)) (2048 (120242 + 2113569 z + 7203382 z^2 - 2319635 z^3 - 21105 z^4 + 560 z^5) EllipticE[1/2 - Sqrt[1 - z]/2]^2)) - (1/(420217875 Pi^2 z^3)) (2048 Sqrt[1 - z] (-92522 - 1462757 z - 4595520 z^2 + 638015 z^3 + 140 z^4) EllipticE[1/2 - Sqrt[1 - z]/2] EllipticK[1/2 - Sqrt[1 - z]/2]) + (1/(420217875 Pi^2 z^3)) (2048 (120242 + 2113569 z + 7203382 z^2 - 2319635 z^3 - 21105 z^4 + 560 z^5) EllipticE[1/2 - Sqrt[1 - z]/2] EllipticK[1/2 - Sqrt[1 - z]/2]) + (1/(420217875 Pi^2 z^3)) (1024 Sqrt[1 - z] (-92522 - 1462757 z - 4595520 z^2 + 638015 z^3 + 140 z^4) EllipticK[1/2 - Sqrt[1 - z]/2]^2) - (1/(420217875 Pi^2 z^3)) (1024 (92522 + 1423426 z + 4025036 z^2 - 3946325 z^3 - 10570 z^4 + 280 z^5) EllipticK[1/2 - Sqrt[1 - z]/2]^2)

Standard Form

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MathML Form

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</apply> </apply> <apply> <times /> <cn type='integer'> 2113569 </cn> <ci> z </ci> </apply> <cn type='integer'> 120242 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <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> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1024 </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'> 140 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 638015 </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'> 4595520 </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'> 1462757 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -92522 </cn> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <plus /> 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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]