Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6h6)

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=-9/8

http://functions.wolfram.com/07.23.03.b6h6.01

Input Form

Hypergeometric2F1[-(45/8), -(9/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1371242496 + 22202344320 z - 181720109285 z^2 + 1074025154930 z^3 - 6399905468775 z^4 - 180550121958780 z^5 - 150406465330275 z^6 - 367490427150 z^7 + 53275874575 z^8 - 6064597000 z^9 + 360655680 z^10) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-1371242496 + 22202344320 z - 181720109285 z^2 + 1074025154930 z^3 - 6399905468775 z^4 - 180550121958780 z^5 - 150406465330275 z^6 - 367490427150 z^7 + 53275874575 z^8 - 6064597000 z^9 + 360655680 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1371242496 + 22202344320 z - 181720109285 z^2 + 1074025154930 z^3 - 6399905468775 z^4 - 180550121958780 z^5 - 150406465330275 z^6 - 367490427150 z^7 + 53275874575 z^8 - 6064597000 z^9 + 360655680 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-1371242496 + 22716560256 z - 189905382485 z^2 + 1139968285235 z^3 - 6785192053185 z^4 - 32316250243305 z^5 + 23934316572705 z^6 + 15564220650825 z^7 - 1569570030875 z^8 + 224573905325 z^9 - 24949644720 z^10 + 1442622720 z^11) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (38255956780452185775 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]