Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6e7)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-45/8, b>=a

For fixed z and a=-45/8, b=-17/8

http://functions.wolfram.com/07.23.03.b6e7.01

Input Form

Hypergeometric2F1[-(45/8), -(17/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (457080832 - 8614902400 z + 84337258135 z^2 - 619845505695 z^3 + 4877815139015 z^4 + 211223253092985 z^5 + 350881694010925 z^6 + 107007671846075 z^7 + 85324317925 z^8 - 5786859925 z^9 + 244730640 z^10) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (457080832 - 8614902400 z + 84337258135 z^2 - 619845505695 z^3 + 4877815139015 z^4 + 211223253092985 z^5 + 350881694010925 z^6 + 107007671846075 z^7 + 85324317925 z^8 - 5786859925 z^9 + 244730640 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (457080832 - 8614902400 z + 84337258135 z^2 - 619845505695 z^3 + 4877815139015 z^4 + 211223253092985 z^5 + 350881694010925 z^6 + 107007671846075 z^7 + 85324317925 z^8 - 5786859925 z^9 + 244730640 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (457080832 - 8786307712 z + 87520977895 z^2 - 650613512445 z^3 + 5102062331795 z^4 + 46349627158695 z^5 - 210978541235 z^6 - 43686971071975 z^7 - 7311965189375 z^8 + 352284657725 z^9 - 23616506760 z^10 + 978922560 z^11) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (42756657578152442925 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

Standard Form

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MathML Form

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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,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]