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/07.34.03.0016.01

 Input Form

 MeijerG[{{0, (Subscript[a, 1] - m)/n, (Subscript[a, 1] - m + 1)/n, \[Ellipsis], (Subscript[a, 1] - m + n - 1)/n, \[Ellipsis], (Subscript[a, p] - m)/n, (Subscript[a, p] - m + 1)/n, \[Ellipsis], (Subscript[a, p] - m + n - 1)/n}, {}}, {{0}, {-(m/n), (-m + 1)/n, \[Ellipsis], (-m + n - 1)/n, (Subscript[b, 1] - m)/n, (Subscript[b, 1] - m + 1)/n, \[Ellipsis], (Subscript[b, 1] - m + n - 1)/n, \[Ellipsis], (Subscript[b, q] - m)/n, (Subscript[b, q] - m + 1)/n, \[Ellipsis], (Subscript[b, q] - m + n - 1)/ n}}, z] == ((2 Pi)^((q - p + 1) ((1 - n)/2)) n^((1/2) (-1 - p + q) + Sum[Subscript[a, j], {j, 1, p}] - Sum[Subscript[b, j], {j, 1, q}]) (Product[Gamma[1 - Subscript[a, j]], {j, 1, p}]/Product[Gamma[1 - Subscript[b, j]], {j, 1, q}]) Sum[Exp[-((2 Pi I k m)/n)] HypergeometricPFQ[{1 - Subscript[a, 1], \[Ellipsis], 1 - Subscript[a, p]}, {1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, q]}, n^(q - p + 1) Exp[(2 Pi I k)/n] (-z)^(1/n)], {k, 0, n - 1}])/(-z)^(m/n) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 G p n , q n + n 1 , p n ( z 0 , a 1 - m n , a 1 - m + 1 n , , a 1 - m + n - 1 n , , a p - m n , a p - m + 1 n , , a p - m + n - 1 n 0 , - m n , - m + 1 n , , - m + n - 1 n , b 1 - m n , b 1 - m + 1 n , , b 1 - m + n - 1 n , , b q - m n , b q - m + 1 n , , b q - m + n - 1 n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["p", " ", "n"]], ",", RowBox[List[RowBox[List["q", " ", "n"]], "+", "n"]]]], RowBox[List["1", ",", RowBox[List["p", " ", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["m", "n"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] ( 2 π ) 1 2 ( q - p + 1 ) ( 1 - n ) j = 1 q Γ ( 1 - b j ) n 1 2 ( q - p - 1 ) + j = 1 p a j - j = 1 q b j ( j = 1 p Γ ( 1 - a j ) ) ( - z ) - m n k = 0 n - 1 - 2 π k m n p F q ( 1 - a 1 , , 1 - a p ; 1 - b 1 , , 1 - b q ; n q - p + 1 2 π k n ( - z ) 1 / n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[RowBox[List[SuperscriptBox["n", RowBox[List["q", "-", "p", "+", "1"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "n"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "/", "n"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; m + n + G p n , q n + n 1 , p n ( z 0 , a 1 - m n , a 1 - m + 1 n , , a 1 - m + n - 1 n , , a p - m n , a p - m + 1 n , , a p - m + n - 1 n 0 , - m n , - m + 1 n , , - m + n - 1 n , b 1 - m n , b 1 - m + 1 n , , b 1 - m + n - 1 n , , b q - m n , b q - m + 1 n , , b q - m + n - 1 n ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["p", " ", "n"]], ",", RowBox[List[RowBox[List["q", " ", "n"]], "+", "n"]]]], RowBox[List["1", ",", RowBox[List["p", " ", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["m", "n"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] ( 2 π ) 1 2 ( q - p + 1 ) ( 1 - n ) j = 1 q Γ ( 1 - b j ) n 1 2 ( q - p - 1 ) + j = 1 p a j - j = 1 q b j ( j = 1 p Γ ( 1 - a j ) ) ( - z ) - m n k = 0 n - 1 - 2 π k m n p F q ( 1 - a 1 , , 1 - a p ; 1 - b 1 , , 1 - b q ; n q - p + 1 2 π k n ( - z ) 1 / n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[RowBox[List[SuperscriptBox["n", RowBox[List["q", "-", "p", "+", "1"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "n"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "/", "n"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; m + n + [/itex]

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

 2001-10-29