Generalized hypergeometric function 2F3: Identities (formula 07.26.17.0002)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Identities

Recurrence identities

Consecutive neighbors

http://functions.wolfram.com/07.26.17.0002.01

Input Form

HypergeometricPFQ[{a, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == Subscript[B, 1] HypergeometricPFQ[{a - 1, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] + (Subscript[B, 2] + Subscript[C, 2] z) HypergeometricPFQ[ {a - 2, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] + (Subscript[B, 3] + Subscript[C, 3] z) HypergeometricPFQ[{a - 3, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] + Subscript[B, 4] HypergeometricPFQ[{a - 4, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] /; Subscript[B, 1] == (7 - 4 a + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3])/(1 - a) && Subscript[B, 2] == ((3 a - 7) (Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3]) - Subscript[b, 2] Subscript[b, 3] - Subscript[b, 1] Subscript[b, 2] - Subscript[b, 1] Subscript[b, 3] - 31 + 27 a - 6 a^2)/ ((a - 1) (a - 2)) && Subscript[C, 2] == 1/((a - 1) (a - 2)) && Subscript[B, 3] == ((a - 3) (28 - 21 a + 4 a^2 + (8 - 3 a) (Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3]) + 2 (Subscript[b, 2] Subscript[b, 3] + Subscript[b, 1] Subscript[b, 2] + Subscript[b, 1] Subscript[b, 3])) - Subscript[b, 1] Subscript[b, 2] Subscript[b, 3])/((a - 1) (a - 2) (a - 3)) && Subscript[C, 3] == (3 - a + Subscript[a, 2])/((a - 1) (a - 2) (a - 3)) && Subscript[B, 4] == -(((a - Subscript[b, 1] - 3) (a - Subscript[b, 2] - 3) (a - Subscript[b, 3] - 3))/((a - 1) (a - 2) (a - 3)))

Standard Form

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MathML Form

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Rule Form

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SubscriptBox["bb", "2"], " ", SubscriptBox["bb", "3"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "3"]], ")"]]]]]]], "&&", RowBox[List[SubscriptBox["C", "3"], "\[Equal]", FractionBox[RowBox[List["3", "-", "a", "+", SubscriptBox["aa", "2"]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "3"]], ")"]]]]]]], "&&", RowBox[List[SubscriptBox["B", "4"], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["bb", "1"], "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", SubscriptBox["bb", "2"], "-", "3"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", SubscriptBox["bb", "3"], "-", "3"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "3"]], ")"]]]]]]]]]]]]]]]]]

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,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]