Generalized hypergeometric function 3F2: Series representations (formula 07.27.06.0015)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Series representations

Generalized power series

Expansions at z==1

For the function itself

The major terms in the general formula for expansions at z==1

http://functions.wolfram.com/07.27.06.0015.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] ((Gamma[Subscript[\[Psi], 2]] Product[Gamma[Subscript[b, k]], {k, 1, 2}])/ Product[Gamma[Subscript[a, k]], {k, 3, 3}]) Sum[((Pochhammer[Subscript[\[Psi], 2], k] HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, k])/(Gamma[Subscript[\[Psi], 2] + Subscript[a, 1] + k] Gamma[Subscript[\[Psi], 2] + Subscript[a, 2] + k])) (1 + O[z - 1]), {k, 0, Infinity}] + ((Gamma[-Subscript[\[Psi], 2]] Product[Gamma[Subscript[b, k]], {k, 1, 2}])/ Product[Gamma[Subscript[a, k]], {k, 1, 3}]) (1 - z)^Subscript[\[Psi], 2] (1 + O[z - 1]) /; (z -> 1) && Subscript[\[Psi], 2] == Subscript[b, 1] + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] && Re[Subscript[\[Psi], 2]] > 0 && Re[Subscript[a, 3]] > 0

Standard Form

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MathML Form

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</mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <msub> <mi> ψ </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

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,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]