Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8sq)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-19/8

http://functions.wolfram.com/07.23.03.b8sq.01

Input Form

Hypergeometric2F1[-(41/8), -(19/8), 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (73891840 - 1375946880 z + 13310496045 z^2 - 96740629755 z^3 + 754259027445 z^4 + 36567779540481 z^5 + 63222249317283 z^6 + 21773334326463 z^7 + 695385202743 z^8 - 27714748677 z^9 + 935331012 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 10 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (4618240 - 85239000 z + 818152335 z^2 - 5916202215 z^3 + 2555255958255 z^4 + 6010623720801 z^5 + 3313468518885 z^6 + 418207032099 z^7 - 2304922851 z^8 + 77944251 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-9236480 + 165282480 z - 1544375085 z^2 + 10981827450 z^3 + 8277082012485 z^4 + 16123234568328 z^5 + 6094046525517 z^6 + 230291857026 z^7 - 9186286725 z^8 + 311777004 z^9) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (73891840 - 1375946880 z + 13310496045 z^2 - 96740629755 z^3 + 754259027445 z^4 + 36567779540481 z^5 + 63222249317283 z^6 + 21773334326463 z^7 + 695385202743 z^8 - 27714748677 z^9 + 935331012 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (6577315213128885225 Pi 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]