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 MeijerG

 http://functions.wolfram.com/07.34.04.0014.01

 Input Form

 Limit[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]}}, x - I \[Epsilon]], \[Epsilon] -> Plus[0]] == Sum[(Product[If[j == k, 1, Gamma[Subscript[a, k] - Subscript[a, j]]] Product[Gamma[1 - Subscript[a, k] + Subscript[b, j]], {j, 1, m}], {j, 1, n}]/Product[Gamma[Subscript[a, k] - Subscript[b, j]] Product[Gamma[1 + Subscript[a, j] - Subscript[a, k]], {j, n + 1, p}], {j, m + 1, q}]) x^(Subscript[a, k] - 1) Exp[-2 Subscript[a, k] Pi I] HypergeometricPFQ[{1 - Subscript[a, k] + Subscript[b, 1], \[Ellipsis], 1 - Subscript[a, k] + Subscript[b, q]}, {1 + Subscript[a, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, k - 1] - Subscript[a, k], 1 + Subscript[a, k + 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[a, k]}, (-1)^(q - m - n)/x], {k, 1, n}] /; (p > q || (p == q && m + n == p + 1 && !IntervalMemberQ[Interval[{-1, 0}], x]) || (p == q && m + n > p + 1) || (p == q && m + n == p && Abs[x] > 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", 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"]]], "}"]]]], "}"]], ",", RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "j"]]], "]"]]]], "]"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "j"]]], "]"]]]], ")"]], " "]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["m", "+", "1"]]]], "q"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "j"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "p"], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]]]], SuperscriptBox["x", RowBox[List[SubscriptBox["a", "k"], "-", "1"]]], RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "2"]], SubscriptBox["a", "k"], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["a", "k"]]], ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["a", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "p"], "-", SubscriptBox["a", "k"]]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n"]]], " "]], "x"]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["p", ">", "q"]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["m", "+", "n"]], "\[Equal]", RowBox[List["p", "+", "1"]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", " ", "x"]], "]"]], "]"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["m", "+", "n"]], ">", RowBox[List["p", "+", "1"]]]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["m", "+", "n"]], "\[Equal]", "p"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "x", "]"]], ">", "1"]]]], ")"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "n"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "n"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]], ")"]]]], ")"]]]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 lim ϵ "\[Rule]" + 0 TagBox["", HypergeometricPFQ] G TagBox["G", MeijerG] p , q m , n ( x - ϵ TagBox[RowBox[List["x", "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], 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 = 1 n ( j = 1 j k n Γ ( a k - a j ) ) j = 1 m Γ ( 1 - a k + b j ) j = m + 1 q Γ ( a k - b j ) j = n + 1 p Γ ( 1 - a k + a j ) x a k - 1 - 2 a k π q F p - 1 ( 1 - a k + b 1 , , 1 - a k + b q TagBox[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; 1 - a k + a 1 TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]] , TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]] , 1 - a k + a k - 1 TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]] , 1 - a k + a k + 1 TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]] , TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]] , 1 - a k + a p TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", "p"]]], HypergeometricPFQ, Rule[Editable, True]] ; ( - 1 ) q - m - n x TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n"]]], "x"], HypergeometricPFQRegularized, Rule[Editable, True]] ) /; ( p > q p q m + n p + 1 x ( - 1 , 0 ) p q m + n > p + 1 p q m + n p "\[LeftBracketingBar]" x "\[RightBracketingBar]" > 1 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j n 1 k n ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) x < 0 FormBox RowBox RowBox RowBox Limit [ RowBox TagBox HypergeometricPFQ RowBox SubsuperscriptBox TagBox G MeijerG RowBox p , q RowBox m , n RowBox ( RowBox TagBox RowBox x - RowBox ϵ 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 ϵ -> RowBox + 0 ] RowBox UnderoverscriptBox RowBox k = 1 n RowBox FractionBox RowBox RowBox ( RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k n RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox a j ) ) RowBox UnderoverscriptBox RowBox j = 1 m RowBox Γ ( RowBox 1 - SubscriptBox a k + SubscriptBox b j ) RowBox UnderoverscriptBox RowBox j = RowBox m + 1 q RowBox RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox b j ) RowBox UnderoverscriptBox RowBox j = RowBox n + 1 p RowBox Γ ( RowBox 1 - SubscriptBox a k + SubscriptBox a j ) SuperscriptBox x RowBox SubscriptBox a k - 1 SuperscriptBox RowBox RowBox - 2 SubscriptBox a k π RowBox RowBox SubscriptBox FormBox q TraditionalForm SubscriptBox F FormBox RowBox p - 1 TraditionalForm RowBox ( RowBox RowBox TagBox TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox a k + SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a k + SubscriptBox b q HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox 1 - SubscriptBox a k + SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a k + SubscriptBox a RowBox k - 1 HypergeometricPFQ Rule Editable , TagBox RowBox 1 - SubscriptBox a k + SubscriptBox a RowBox k + 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , RowBox TagBox RowBox 1 - SubscriptBox a k + SubscriptBox a p HypergeometricPFQ Rule Editable ; TagBox FractionBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox q - m - n x HypergeometricPFQRegularized Rule Editable ) /; RowBox RowBox ( RowBox RowBox p > q RowBox RowBox p q RowBox RowBox m + n RowBox p + 1 RowBox x RowBox ( RowBox RowBox - 1 , 0 ) RowBox RowBox p q RowBox RowBox m + n > RowBox p + 1 RowBox RowBox p q RowBox RowBox m + n p RowBox RowBox x > 1 ) RowBox SubscriptBox RowBox RowBox { RowBox j , k } , RowBox RowBox RowBox { RowBox j , k } TagBox Function RowBox j k RowBox 1 j n RowBox 1 k n RowBox ( RowBox RowBox SubscriptBox a j - SubscriptBox a k TagBox Function ) RowBox x < 0 TraditionalForm [/itex]

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

 2007-05-02