Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8xo)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-21/4, b>=a

For fixed z and a=-21/4, b=21/4

http://functions.wolfram.com/07.23.03.a8xo.01

Input Form

Hypergeometric2F1[-(21/4), 21/4, 6, -z] == (1/(168441498225 Pi z^5)) (16384 (1 + z)^(1/4) (2 (43008 - 35616 z + 62671 z^2 - 180075 z^3 + 1008105 z^4 + 48536191 z^5 + 190547148 z^6 + 321282192 z^7 + 279910400 z^8 + 124481280 z^9 + 22471680 z^10) EllipticE[1/2 - 1/(2 Sqrt[1 + z])] - Sqrt[1 + z] (43008 - 67872 z + 118615 z^2 - 278880 z^3 + 1235115 z^4 + 27008146 z^5 + 77281776 z^6 + 92732640 z^7 + 51941120 z^8 + 11235840 z^9) EllipticK[1/2 - 1/(2 Sqrt[1 + z])] - (43008 - 35616 z + 62671 z^2 - 180075 z^3 + 1008105 z^4 + 48536191 z^5 + 190547148 z^6 + 321282192 z^7 + 279910400 z^8 + 124481280 z^9 + 22471680 z^10) EllipticK[1/2 - 1/(2 Sqrt[1 + z])]))

Standard Form

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MathML Form

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</cn> <apply> <times /> <cn type='integer'> 180075 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 62671 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 35616 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 43008 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},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]