Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9uf)

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=-19/4, b>=a

For fixed z and a=-19/4, b=21/4

http://functions.wolfram.com/07.23.03.a9uf.01

Input Form

Hypergeometric2F1[-(19/4), 21/4, -(7/2), z] == (1/(53040 Pi^(3/2))) ((-((1/(-1 + z)^4) (8 Sqrt[z] (-6630 - 23205 z - 81549 z^2 - 385203 z^3 + 72187755 z^4 - 289743360 z^5 + 440330240 z^6 - 297795584 z^7 + 75497472 z^8) EllipticE[(1/2) (1 - Sqrt[z])])) + (1/(-1 + z)^4) (8 Sqrt[z] (-6630 - 23205 z - 81549 z^2 - 385203 z^3 + 72187755 z^4 - 289743360 z^5 + 440330240 z^6 - 297795584 z^7 + 75497472 z^8) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^4 (1 + Sqrt[z])^3)) ((53040 - 79560 Sqrt[z] + 225420 z - 318240 z^(3/2) + 825435 z^2 - 1151631 z^(5/2) + 3720093 z^3 - 5260905 z^(7/2) + 38520300 z^4 + 250230720 z^(9/2) - 453857280 z^5 - 705116160 z^(11/2) + 1073192960 z^6 + 688128000 z^(13/2) - 964689920 z^7 - 226492416 z^(15/2) + 301989888 z^8) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^3 (1 + Sqrt[z])^4)) ((53040 + 79560 Sqrt[z] + 225420 z + 318240 z^(3/2) + 825435 z^2 + 1151631 z^(5/2) + 3720093 z^3 + 5260905 z^(7/2) + 38520300 z^4 - 250230720 z^(9/2) - 453857280 z^5 + 705116160 z^(11/2) + 1073192960 z^6 - 688128000 z^(13/2) - 964689920 z^7 + 226492416 z^(15/2) + 301989888 z^8) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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</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]