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 MeijerG

 http://functions.wolfram.com/07.34.06.0052.01

 Input Form

 MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z] == Sum[(1/k!) MeijerG[{{1 - Subscript[b, 1] + k, \[Ellipsis], 1 - Subscript[b, m] + k}, {1, 1 - Subscript[b, m + 1] + k, \[Ellipsis], 1 - Subscript[b, q] + k}}, {{1 + k, 1 - Subscript[a, 1] + k, \[Ellipsis], 1 - Subscript[a, n] + k}, {1 - Subscript[a, n + 1] + k, \[Ellipsis], 1 - Subscript[a, p] + k}}, 1/x/E^(2 Pi I Floor[Arg[z - x]/(2 Pi)])] (z - x)^k, {k, 0, Infinity}] /; Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 0 && Element[p, Integers] && p >= 0 && Element[q, Integers] && q >= 0 && m <= q && n <= p && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "m"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "m"], "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"], "+", "k"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List["1", "-", SubscriptBox["a", "1"], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "n"], "+", "k"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "p"], "+", "k"]]]], "}"]]]], "}"]], ",", " ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], FractionBox["1", "x"]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", " ", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "q"]], "\[And]", RowBox[List["n", "\[LessEqual]", "p"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 G TagBox["G", MeijerG] p , q m , n ( z TagBox["z", MeijerG, Rule[Editable, True]] a 1 TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , a n TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]] , a n + 1 TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , a p TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]] b 1 TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , b m TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]] , b m + 1 TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]] , TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] , b q TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]] ) k = 0 1 k ! G p + 1 , q + 1 m , n + 1 ( - 2 π arg ( z - x ) 2 π 1 x 1 - b 1 + k , , 1 - b m + k , 1 , 1 - b m + 1 + k , , 1 - b q + k 1 + k , 1 - a 1 + k , , 1 - a n + k , 1 - a n + 1 + k , , 1 - a p + k ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["p", "+", "1"]], ",", RowBox[List["q", "+", "1"]]]], RowBox[List["m", ",", RowBox[List["n", "+", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List[TagBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["arg", "(", RowBox[List["z", "-", "x"]], ")"]], RowBox[List["2", " ", "\[Pi]"]]], "\[RightFloor]"]]]]], MeijerG, Rule[Editable, True]], FractionBox["1", "x"]]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "m"], "+", "k"]], ",", "1", ",", RowBox[List["1", "-", SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"], "+", "k"]]]]], List[RowBox[List[RowBox[List["1", "+", "k"]], ",", RowBox[List["1", "-", SubscriptBox["a", "1"], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "n"], "+", "k"]], ",", RowBox[List["1", "-", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "p"], "+", "k"]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] ( z - x ) k /; m n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] p q TagBox["\[DoubleStruckCapitalN]", Function[Integers]] m q n p x < 0 FormBox RowBox RowBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox p , q RowBox m , n RowBox ( RowBox TagBox z MeijerG Rule Editable GridBox RowBox TagBox SubscriptBox a 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox a n MeijerG Rule Editable , TagBox SubscriptBox a RowBox n + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox a p MeijerG Rule Editable RowBox TagBox SubscriptBox b 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox b m MeijerG Rule Editable , TagBox SubscriptBox b RowBox m + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox SubscriptBox b q MeijerG Rule Editable ) RowBox UnderoverscriptBox RowBox k = 0 RowBox FractionBox 1 RowBox k ! TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox RowBox p + 1 , RowBox q + 1 RowBox m , RowBox n + 1 RowBox ( RowBox RowBox TagBox SuperscriptBox RowBox RowBox - 2 π RowBox FractionBox RowBox arg ( RowBox z - x ) RowBox 2 π MeijerG Rule Editable FractionBox 1 x GridBox RowBox RowBox 1 - SubscriptBox b 1 + k , , RowBox 1 - SubscriptBox b m + k , 1 , RowBox 1 - SubscriptBox b RowBox m + 1 + k , , RowBox 1 - SubscriptBox b q + k RowBox RowBox 1 + k , RowBox 1 - SubscriptBox a 1 + k , , RowBox 1 - SubscriptBox a n + k , RowBox 1 - SubscriptBox a RowBox n + 1 + k , , RowBox 1 - SubscriptBox a p + k ) MeijerG Rule Editable SuperscriptBox RowBox ( RowBox z - x ) k /; RowBox RowBox m RowBox n TagBox Function RowBox p RowBox q TagBox Function RowBox m q RowBox n p RowBox x < 0 TraditionalForm [/itex]

 Rule Form

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

 2007-05-02