html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 MeijerG

 http://functions.wolfram.com/07.34.06.0056.01

 Input Form

 MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, q]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z] == Pi^(n - 1) Sum[(Product[Gamma[1 + Subscript[b, j] - Subscript[a, k]], {j, 1, m}]/Product[If[j == k, 1, Sin[Pi (Subscript[a, k] - Subscript[a, j])]] Product[Gamma[Subscript[a, k] - Subscript[b, j]], {j, m + 1, q}], {j, 1, n}]) E^(Pi I (2 Subscript[a, k] Floor[Arg[z + 1]/(2 Pi)] + Subscript[a, k] - 1) (q - m - n - 2 Floor[(q - m - n)/2])) Sum[(((-1)^v Pochhammer[1 - Subscript[a, k], v])/v!) ((-1)^(m + n - q) z - 1)^v, {v, 0, Infinity}] HypergeometricPFQRegularized[{1 + Subscript[b, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[b, q] - Subscript[a, k]}, {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)/z], {k, 1, n}] /; ((m + n == q + 1 && !IntervalMemberQ[Interval[{-1, 0}], z]) || m + n > q + 1 || (m + n == q && Abs[z] > 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

 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", "q"]]], "}"]]]], "}"]], ",", 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[SuperscriptBox["\[Pi]", RowBox[List["n", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Sin", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "j"]]], ")"]]]], "]"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["m", "+", "1"]]]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["b", "j"]]], "]"]]]]]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["a", "k"], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", SubscriptBox["a", "k"], "-", "1"]], ")"]], RowBox[List["(", RowBox[List["q", "-", "m", "-", "n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["q", "-", "m", "-", "n"]], "2"], "]"]]]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["v", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "v"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "k"]]], ",", "v"]], "]"]]]], RowBox[List["v", "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n", "-", "q"]]], "z"]], "-", "1"]], ")"]], "v"]]]]], ")"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["a", "k"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["a", "k"]]]]], "}"]], ",", 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"]]], " "]], "z"]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["m", "+", "n"]], "\[Equal]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", " ", "z"]], "]"]], "]"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["m", "+", "n"]], ">", RowBox[List["q", "+", "1"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["m", "+", "n"]], "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "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"]], ")"]]]], ")"]]]]]]]]]]

 MathML Form

 G q , q m , n ( z a 1 , , a n , a n + 1 , , a q b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["q", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "q"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] π n - 1 k = 1 n j = 1 m Γ ( 1 - a k + b j ) j = 1 j k n sin ( π ( a k - a j ) ) j = m + 1 q Γ ( a k - b j ) π ( 2 arg ( z + 1 ) 2 π a k + a k - 1 ) ( q - m - n - 2 1 2 ( q - m - n ) ) ( v = 0 ( - 1 ) v ( 1 - a k ) v TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "k"]]], ")"]], "v"], Pochhammer] v ! ( ( - 1 ) m + n - q z - 1 ) v ) q F ~ p - 1 ( 1 - a k + b 1 , , 1 - a k + b q ; 1 + a 1 - a k , , 1 + a k - 1 - a k , 1 + a k + 1 - a k , , 1 + a p - a k ; ( - 1 ) q - m - n z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["q", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["p", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List["1", "+", SubscriptBox["a", "p"], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n"]]], "z"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] /; ( m + n q + 1 z ( - 1 , 0 ) m + n > q + 1 m + n q "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] j k 1 j n 1 k n ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] ) G q , q m , n ( z a 1 , , a n , a n + 1 , , a q b 1 , , b m , b m + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["q", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "q"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] π n - 1 k = 1 n j = 1 m Γ ( 1 - a k + b j ) j = 1 j k n sin ( π ( a k - a j ) ) j = m + 1 q Γ ( a k - b j ) π ( 2 arg ( z + 1 ) 2 π a k + a k - 1 ) ( q - m - n - 2 1 2 ( q - m - n ) ) ( v = 0 ( - 1 ) v ( 1 - a k ) v TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "k"]]], ")"]], "v"], Pochhammer] v ! ( ( - 1 ) m + n - q z - 1 ) v ) q F ~ p - 1 ( 1 - a k + b 1 , , 1 - a k + b q ; 1 + a 1 - a k , , 1 + a k - 1 - a k , 1 + a k + 1 - a k , , 1 + a p - a k ; ( - 1 ) q - m - n z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["q", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["p", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List["1", "+", SubscriptBox["a", "p"], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n"]]], "z"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] /; ( m + n q + 1 z ( - 1 , 0 ) m + n > q + 1 m + n q "\[LeftBracketingBar]" z "\[RightBracketingBar]" > 1 ) { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] j k 1 j n 1 k n ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] ) [/itex]

 Rule Form

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

 2007-05-02

© 1998-2013 Wolfram Research, Inc.