Generalized hypergeometric function 3F2: Specific values (formula 07.27.03.alcd)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]

Specific values

For integer and half-integer parameters and fixed z

For fixed z and a₁=-1/2, a₂>=-1/2

For fixed z and a₁=-1/2, a₂=2, a₃>=2

For fixed z and a₁=-1/2, a₂=2, a₃=4

For fixed z and a₁=-1/2, a₂=2, a₃=4, b₁=1/2

http://functions.wolfram.com/07.27.03.alcd.01

Input Form

HypergeometricPFQ[{-(1/2), 2, 4}, {1/2, 3/2}, -z] == (708 + 315 Pi^2 Sqrt[z] + 3068 z + 945 Pi^2 z^(3/2) + 3500 z^2 + 945 Pi^2 z^(5/2) + 1260 z^3 + 315 Pi^2 z^(7/2))/(768 (1 + z)^3) + ((-128 - 1646 z - 4578 z^2 - 5632 z^3 - 3274 z^4 - 737 z^5) Log[1 + Sqrt[z] - Sqrt[1 + z]])/(128 (1 + z)^(9/2)) + ((128 + 1646 z + 4578 z^2 + 5632 z^3 + 3274 z^4 + 737 z^5) Log[1 - Sqrt[z] + Sqrt[1 + z]])/(128 (1 + z)^(9/2)) + ((5 + 176 z + 406 z^2 + 350 z^3 + 105 z^4) Log[Sqrt[z] + Sqrt[1 + z]])/ (64 Sqrt[z] (1 + z)^(7/2)) + ((128 + 1646 z + 4578 z^2 + 5632 z^3 + 3274 z^4 + 737 z^5) Log[(-1 + Sqrt[z] + Sqrt[1 + z])/(1 + Sqrt[z] + Sqrt[1 + z])])/ (128 (1 + z)^(9/2)) + (105/64) Sqrt[z] Log[Sqrt[z] + Sqrt[1 + z]] Log[(-1 + Sqrt[z] + Sqrt[1 + z])/(1 + Sqrt[z] + Sqrt[1 + z])] + (105/64) Sqrt[z] PolyLog[2, -(1/(Sqrt[z] + Sqrt[1 + z]))] - (105/64) Sqrt[z] PolyLog[2, 1/(Sqrt[z] + Sqrt[1 + z])]

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},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},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]