Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9095)

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=-23/4, b>=a

For fixed z and a=-23/4, b=-17/4

http://functions.wolfram.com/07.23.03.9095.01

Input Form

Hypergeometric2F1[-(23/4), -(17/4), 6, -z] == -((16384 (1 + z)^(1/4) (2 (15214592 + 375372512 z + 5001225087 z^2 + 52697819512 z^3 + 639635859940 z^4 - 29672591327736 z^5 + 93576203494074 z^6 - 82829382505656 z^7 + 22721048653284 z^8 - 1506334400040 z^9 + 302344575 z^10) EllipticE[1/2 - 1/(2 Sqrt[1 + z])] + (-15214592 (1 + Sqrt[1 + z]) + 302344575 z^10 (-1 + 2 Sqrt[1 + z]) - 237728 z (1579 + 1531 Sqrt[1 + z]) - 2860165 z^4 (223636 + 210925 Sqrt[1 + z]) - 22287 z^2 (224401 + 212233 Sqrt[1 + z]) - 7429 z^3 (7093528 + 6621655 Sqrt[1 + z]) + 15 z^9 (100422293336 + 4938294725 Sqrt[1 + z]) - 27 z^8 (841520320492 + 102728972865 Sqrt[1 + z]) + 21 z^5 (1412980539416 + 557849633501 Sqrt[1 + z]) - 21 z^6 (4456009690194 + 1253338032271 Sqrt[1 + z]) + 3 z^7 (27609794168552 + 5427932720141 Sqrt[1 + z])) EllipticK[1/2 - 1/(2 Sqrt[1 + z])]))/ (150770847054002175 Pi z^5))

Standard Form

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MathML Form

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</apply> </apply> <cn type='integer'> 224401 </cn> </apply> <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'> 237728 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 1531 </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'> 1579 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15214592 </cn> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </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]