Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b87p)

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=-43/8, b>=a

For fixed z and a=-43/8, b=39/8

http://functions.wolfram.com/07.23.03.b87p.01

Input Form

Hypergeometric2F1[-(43/8), 39/8, 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-2650374144 + 735064704 z + 1167060807 z^2 + 3118444065 z^3 + 16454918403 z^4 - 363522541205 z^5 + 1203567730330 z^6 - 1807698074960 z^7 + 1437887349600 z^8 - 592597674240 z^9 + 100164979200 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (165648384 + 74412360 z - 7198587 z^2 - 189185337 z^3 - 108526787325 z^4 + 442106728505 z^5 - 743842357600 z^6 + 640988700560 z^7 - 281274090240 z^8 + 50082489600 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (331296768 - 37529712 z - 135479025 z^2 - 405304713 z^3 + 126714138705 z^4 - 588055710755 z^5 + 1182425357260 z^6 - 1300257654800 z^7 + 819137879040 z^8 - 279108360960 z^9 + 40065991680 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-2650374144 + 735064704 z + 1167060807 z^2 + 3118444065 z^3 + 16454918403 z^4 - 363522541205 z^5 + 1203567730330 z^6 - 1807698074960 z^7 + 1437887349600 z^8 - 592597674240 z^9 + 100164979200 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (168872880962864925 Pi z^5)

Standard Form

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MathML Form

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</msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 100164979200 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 592597674240 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1437887349600 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1807698074960 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1203567730330 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 363522541205 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16454918403 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3118444065 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 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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]