Generalized hypergeometric function 4F3: Identities (formula 07.28.17.0003)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Identities

Recurrence identities

Consecutive neighbors

http://functions.wolfram.com/07.28.17.0003.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {b, Subscript[b, 2], Subscript[b, 3]}, z] == ((Subscript[B, 1] + Subscript[C, 1] z)/(z - 1)) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {1 + b, Subscript[b, 2], Subscript[b, 3]}, z] + ((Subscript[B, 2] + Subscript[C, 2] z)/(z - 1)) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {2 + b, Subscript[b, 2], Subscript[b, 3]}, z] + ((Subscript[B, 3] + Subscript[C, 3] z)/(z - 1)) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {3 + b, Subscript[b, 2], Subscript[b, 3]}, z] + ((Subscript[C, 4] z)/(z - 1)) HypergeometricPFQ[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {4 + b, Subscript[b, 2], Subscript[b, 3]}, z] /; Subscript[B, 1] == (Subscript[b, 2] + Subscript[b, 3] - 3 b - 5)/b && Subscript[C, 1] == (6 - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] - Subscript[a, 4] + 4 b)/b && Subscript[B, 2] == ((2 + b) (7 + 3 b - 2 Subscript[b, 2] - 2 Subscript[b, 3]) + Subscript[b, 2] Subscript[b, 3])/(b (b + 1)) && Subscript[C, 2] == (1/(b (1 + b))) (3 (2 + b) (Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4]) - Subscript[a, 1] Subscript[a, 4] - Subscript[a, 2] Subscript[a, 4] - Subscript[a, 3] Subscript[a, 4] - Subscript[a, 1] Subscript[a, 2] - Subscript[a, 1] Subscript[a, 3] - Subscript[a, 2] Subscript[a, 3] - 25 - 24 b - 6 b^2) && Subscript[B, 3] == -(((3 + b - Subscript[b, 2]) (3 + b - Subscript[b, 3]))/ (b (1 + b))) && Subscript[C, 3] == (1/(b (1 + b) (2 + b))) ((5 + 2 b) (13 + 10 b + 2 b^2 + Subscript[a, 1] Subscript[a, 3] + Subscript[a, 2] Subscript[a, 3] + Subscript[a, 1] Subscript[a, 4] + Subscript[a, 2] Subscript[a, 4] + Subscript[a, 1] Subscript[a, 2] + Subscript[a, 3] Subscript[a, 4]) - (19 + 15 b + 3 b^2) (Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + Subscript[a, 4]) - Subscript[a, 1] Subscript[a, 3] Subscript[a, 4] - Subscript[a, 2] Subscript[a, 3] Subscript[a, 4] - Subscript[a, 1] Subscript[a, 2] Subscript[a, 4] - Subscript[a, 1] Subscript[a, 2] Subscript[a, 3]) && Subscript[C, 4] == -(((3 - Subscript[a, 1] + b) (3 - Subscript[a, 2] + b) (3 - Subscript[a, 3] + b) (3 - Subscript[a, 4] + b))/ (b (1 + b) (2 + b) (3 + b)))

Standard Form

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MathML Form

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<mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> </mrow> <mo> + </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ⁢ </mo> <msub> <mi> a </mi> <mn> 4 </mn> 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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]