Generalized hypergeometric function 1F2: Series representations (formula 07.22.06.0012)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁},{b₁,b₂},z]

Series representations

Asymptotic series expansions

Expansions for any z in trigonometric form

http://functions.wolfram.com/07.22.06.0012.01

Input Form

HypergeometricPFQ[{Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] (((-z)^((1/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2]))/ (Sqrt[Pi] Gamma[Subscript[a, 1]])) (Cos[2 Sqrt[-z] + (Pi/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])] (1 + O[1/z]) + (1/(16 Sqrt[-z])) (12 Subscript[a, 1]^2 - 4 Subscript[b, 1]^2 + 8 Subscript[b, 2] - 4 Subscript[b, 2]^2 + 8 Subscript[b, 1] (Subscript[b, 2] + 1) - 8 Subscript[a, 1] (Subscript[b, 1] + Subscript[b, 2] + 1) - 3) Sin[2 Sqrt[-z] + (Pi/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])] (1 + O[1/z])) + (1/((-z)^Subscript[a, 1] (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]]))) (1 + O[1/z])) /; (Abs[z] -> Infinity)

Standard Form

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MathML Form

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</mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> 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</apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </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,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},{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]