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variants of this functions
HypergeometricPFQRegularized




Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQRegularized[{a1,...,ap},{b1,...,bq},z] > Series representations > Generalized power series > Expansions at z==infinity for p==q+1 > Case of poles of order r in the points ar+k/;r ∈ {2,3,4} && k ∈ N





http://functions.wolfram.com/07.32.06.0020.01









  


  








Input Form





HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == AsymptoticHypergeometricPFQRegularizedSeries[Power][ {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, {z, Infinity, Infinity}] /; !IntervalMemberQ[Interval[{0, 1}], z] && Element[Subscript[a, k] - Subscript[a, k - 1], Integers] && Subscript[a, k] - Subscript[a, k - 1] >= 0 && 2 <= k <= r && !Element[Subscript[a, k] - Subscript[a, 1], Integers] && r + 1 <= k <= q + 1 && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && r + 1 <= j <= q + 1 && r + 1 <= k <= q + 1, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && Element[r, {2, 3, 4}]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["AsymptoticHypergeometricPFQRegularizedSeries", "[", "Power", "]"]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]], ",", RowBox[List["{", RowBox[List["z", ",", "\[Infinity]", ",", "\[Infinity]"]], "}"]]]], "]"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", "r"]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["aa", "1"]]], "\[Element]", "Integers"]]]], "&&", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[NotEqual]", "k"]], "&&", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List[RowBox[List["r", "+", "1"]], "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]]]], ")"]]]], "&&", RowBox[List["r", "\[Element]", RowBox[List["{", RowBox[List["2", ",", "3", ",", "4"]], "}"]]]]]]]]]]]]


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





2001-10-29