Regularized generalized hypergeometric function: Series representations (formula 07.32.06.0040)

HypergeometricPFQRegularized

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]

Series representations

Main terms of asymptotic expansions

http://functions.wolfram.com/07.32.06.0040.01

Input Form

HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] \[Proportional] (1/Product[Gamma[Subscript[a, j]], {j, 1, p}]) (Sum[Residue[(Gamma[s] Product[Gamma[Subscript[a, j] - s], {j, 1, p}])/ Product[Gamma[Subscript[b, j] - s], {j, 1, q}]/(-z)^s, {s, Subscript[a, k]}] (1 + O[1/z]), {k, 1, p}] + KroneckerDelta[q, p + 1] Subscript[d, 1] (-z)^\[Chi] Cos[2 Sqrt[-z] + \[Chi] Pi] (1 + O[1/Sqrt[-z]]) + (UnitStep[q - p] - KroneckerDelta[q, p + 1]) Subscript[d, 2] z^\[Chi] Exp[\[Beta] z^(1/\[Beta])] (1 + O[1/z^(1/\[Beta])])) /; (Abs[z] -> Infinity) && \[Beta] == q - p + 1 && \[Chi] == (1/\[Beta]) ((\[Beta] - 1)/2 + Sum[Subscript[a, k], {k, 1, p}] - Sum[Subscript[b, k], {k, 1, q}]) && 2 Subscript[d, 2] == Subscript[d, 1] == (2 (2 Pi)^((1 - \[Beta])/2))/Sqrt[\[Beta]]

Standard Form

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MathML Form

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<mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> d </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⩵ </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> β </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <msqrt> <mi> β </mi> </msqrt> </mfrac> </mrow> </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> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> p </ci> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </list> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </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> <times /> <apply> <ci> KroneckerDelta </ci> <ci> q </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <ci> χ </ci> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <pi /> <ci> χ </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> UnitStep </ci> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <ci> q </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> χ </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> β </ci> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> β </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> β </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <eq /> <ci> β </ci> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <eq /> <ci> χ </ci> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> β </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> β </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> d </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> β </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> β </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,a₂},{b₁},z]