Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a85h)

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.a85h.01

Input Form

Hypergeometric2F1[-(21/4), -(21/4), 6, z] == (16384 (2 (28514304 - 747163872 z + 10634169273 z^2 - 120504827352 z^3 + 1584723786828 z^4 + 120401581743928 z^5 + 467117413896358 z^6 + 525396327335128 z^7 + 199922927611628 z^8 + 23340990842408 z^9 + 529774888169 z^10) EllipticE[(1/2) (1 - Sqrt[1 - z])] - (28514304 (1 + Sqrt[1 - z]) - 445536 (1677 + 1661 Sqrt[1 - z]) z + 4641 (2291353 + 2252209 Sqrt[1 - z]) z^2 - 37128 (3245659 + 3177557 Sqrt[1 - z]) z^3 + 129948 (12195061 + 11977113 Sqrt[1 - z]) z^4 + 8 (15050197717991 + 8878271521373 Sqrt[1 - z]) z^5 + (467117413896358 + 247174276383494 Sqrt[1 - z]) z^6 + 8 (65674540916891 + 31636841338173 Sqrt[1 - z]) z^7 + 4 (49980731902907 + 21793570139711 Sqrt[1 - z]) z^8 + 8 (2917623855301 + 1125688444583 Sqrt[1 - z]) z^9 + (529774888169 + 168441498225 Sqrt[1 - z]) z^10) EllipticK[(1/2) (1 - Sqrt[1 - z])]))/(407563575727683375 Pi z^5)

Standard Form

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MathML Form

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/> <cn type='integer'> 445536 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 1661 </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> <cn type='integer'> 1677 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 28514304 </cn> <apply> <plus /> <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> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </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> </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]