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variants of this functions
HypergeometricPFQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,a2},{b1,b2},z] > Series representations > Asymptotic series expansions > Case of simple poles





http://functions.wolfram.com/07.25.06.0008.01









  


  










Input Form





HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]])) z^\[Chi] E^z Sum[Subscript[c, k]/z^k, {k, 0, Infinity}] + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[a, 2] - Subscript[a, 1]])/(Gamma[Subscript[a, 2]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - 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[a, 2]}, -(1/z)])/ (-z)^Subscript[a, 1] + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - 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[a, 1]}, -(1/z)])/ (-z)^Subscript[a, 2] /; (Abs[z] -> Infinity) && \[Chi] == Subscript[a, 1] + Subscript[a, 2] - Subscript[b, 1] - Subscript[b, 2] && Subscript[c, 0] == 1 && Subscript[c, 1] == (Subscript[A, 2] - 1) \[Chi] + Subscript[b, 1] Subscript[b, 2] - Subscript[a, 1] Subscript[a, 2] && Subscript[c, k] == (1/k) ((1 - Subscript[B, 2] + 2 Subscript[a, 1] + Subscript[a, 1]^2 + 2 Subscript[a, 2] + Subscript[a, 2]^2 - Subscript[A, 2] Subscript[B, 2] + Subscript[a, 1] Subscript[a, 2] + Subscript[b, 1] Subscript[b, 2] + (2 Subscript[B, 2] - 3 (Subscript[A, 2] + 1)) k + 2 k^2) Subscript[c, k - 1] - (k - Subscript[A, 2] + Subscript[b, 1] - 1) (k - Subscript[A, 2] + Subscript[b, 2] - 1) (k - \[Chi] - 2) Subscript[c, k - 2]) && Subscript[A, 2] == Subscript[a, 1] + Subscript[a, 2] && Subscript[B, 2] == Subscript[b, 1] + Subscript[b, 2] && !Element[Subscript[a, 1] - Subscript[a, 2], Integers]










Standard Form





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MathML Form







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Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.