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variants of this functions
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] > Series representations > Generalized power series > Expansions at z==infinity > Case of simple poles





http://functions.wolfram.com/07.28.06.0017.01









  


  










Input Form





HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[a, 4]])) ((((Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]] Gamma[Subscript[a, 4] - Subscript[a, 1]])/ (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]] Gamma[Subscript[b, 3] - Subscript[a, 1]])) HypergeometricPFQ[ {Subscript[a, 1], 1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 1] - Subscript[b, 3]}, {1 + Subscript[a, 1] - Subscript[a, 2], 1 + Subscript[a, 1] - Subscript[a, 3], 1 + Subscript[a, 1] - Subscript[a, 4]}, 1/z])/(-z)^Subscript[a, 1] + (((Gamma[Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]] Gamma[Subscript[a, 4] - Subscript[a, 2]])/ (Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]] Gamma[Subscript[b, 3] - Subscript[a, 2]])) HypergeometricPFQ[ {Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 3]}, {1 + Subscript[a, 2] - Subscript[a, 1], 1 + Subscript[a, 2] - Subscript[a, 3], 1 + Subscript[a, 2] - Subscript[a, 4]}, 1/z])/(-z)^Subscript[a, 2] + (((Gamma[Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]] Gamma[Subscript[a, 4] - Subscript[a, 3]])/ (Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]] Gamma[Subscript[b, 3] - Subscript[a, 3]])) HypergeometricPFQ[ {Subscript[a, 3], 1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 2], 1 + Subscript[a, 3] - Subscript[b, 3]}, {1 + Subscript[a, 3] - Subscript[a, 1], 1 + Subscript[a, 3] - Subscript[a, 2], 1 + Subscript[a, 3] - Subscript[a, 4]}, 1/z])/(-z)^Subscript[a, 3] + (((Gamma[Subscript[a, 4]] Gamma[Subscript[a, 1] - Subscript[a, 4]] Gamma[Subscript[a, 2] - Subscript[a, 4]] Gamma[Subscript[a, 3] - Subscript[a, 4]])/ (Gamma[Subscript[b, 1] - Subscript[a, 4]] Gamma[Subscript[b, 2] - Subscript[a, 4]] Gamma[Subscript[b, 3] - Subscript[a, 4]])) HypergeometricPFQ[ {Subscript[a, 4], 1 + Subscript[a, 4] - Subscript[b, 1], 1 + Subscript[a, 4] - Subscript[b, 2], 1 + Subscript[a, 4] - Subscript[b, 3]}, {1 + Subscript[a, 4] - Subscript[a, 1], 1 + Subscript[a, 4] - Subscript[a, 2], 1 + Subscript[a, 4] - Subscript[a, 3]}, 1/z])/(-z)^Subscript[a, 4]) /; !Element[Subscript[a, 1] - Subscript[a, 2], Integers] && !Element[Subscript[a, 1] - Subscript[a, 3], Integers] && !Element[Subscript[a, 1] - Subscript[a, 4], Integers] && !Element[Subscript[a, 2] - Subscript[a, 3], Integers] && !Element[Subscript[a, 2] - Subscript[a, 4], Integers] && !Element[Subscript[a, 3] - Subscript[a, 4], Integers]










Standard Form





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MathML Form







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HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </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> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <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> <and /> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29