Generalized hypergeometric function 4F3: Identities (formula 07.28.17.0026)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]

Identities

Functional identities

Major general cases

http://functions.wolfram.com/07.28.17.0026.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[a, 4]])) ((((Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]] Gamma[Subscript[a, 4] - Subscript[a, 1]])/ (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]] Gamma[Subscript[b, 3] - Subscript[a, 1]])) HypergeometricPFQ[ {Subscript[a, 1], 1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 1] - Subscript[b, 3]}, {1 + Subscript[a, 1] - Subscript[a, 2], 1 + Subscript[a, 1] - Subscript[a, 3], 1 + Subscript[a, 1] - Subscript[a, 4]}, 1/z])/(-z)^Subscript[a, 1] + (((Gamma[Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]] Gamma[Subscript[a, 4] - Subscript[a, 2]])/ (Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]] Gamma[Subscript[b, 3] - Subscript[a, 2]])) HypergeometricPFQ[ {Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 3]}, {1 + Subscript[a, 2] - Subscript[a, 1], 1 + Subscript[a, 2] - Subscript[a, 3], 1 + Subscript[a, 2] - Subscript[a, 4]}, 1/z])/(-z)^Subscript[a, 2] + (((Gamma[Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]] Gamma[Subscript[a, 4] - Subscript[a, 3]])/ (Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]] Gamma[Subscript[b, 3] - Subscript[a, 3]])) HypergeometricPFQ[ {Subscript[a, 3], 1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 2], 1 + Subscript[a, 3] - Subscript[b, 3]}, {1 + Subscript[a, 3] - Subscript[a, 1], 1 + Subscript[a, 3] - Subscript[a, 2], 1 + Subscript[a, 3] - Subscript[a, 4]}, 1/z])/(-z)^Subscript[a, 3] + (((Gamma[Subscript[a, 4]] Gamma[Subscript[a, 1] - Subscript[a, 4]] Gamma[Subscript[a, 2] - Subscript[a, 4]] Gamma[Subscript[a, 3] - Subscript[a, 4]])/ (Gamma[Subscript[b, 1] - Subscript[a, 4]] Gamma[Subscript[b, 2] - Subscript[a, 4]] Gamma[Subscript[b, 3] - Subscript[a, 4]])) HypergeometricPFQ[ {Subscript[a, 4], 1 + Subscript[a, 4] - Subscript[b, 1], 1 + Subscript[a, 4] - Subscript[b, 2], 1 + Subscript[a, 4] - Subscript[b, 3]}, {1 + Subscript[a, 4] - Subscript[a, 1], 1 + Subscript[a, 4] - Subscript[a, 2], 1 + Subscript[a, 4] - Subscript[a, 3]}, 1/z])/(-z)^Subscript[a, 4]) /; !Element[Subscript[a, 1] - Subscript[a, 2], Integers] && !Element[Subscript[a, 1] - Subscript[a, 3], Integers] && !Element[Subscript[a, 1] - Subscript[a, 4], Integers] && !Element[Subscript[a, 2] - Subscript[a, 3], Integers] && !Element[Subscript[a, 2] - Subscript[a, 4], Integers] && !Element[Subscript[a, 3] - Subscript[a, 4], Integers] && !IntervalMemberQ[Interval[{0, 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)

2001-10-29

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},{b₁,b₂},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]