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/07.34.03.0009.01

 Input Form

 MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {}}, {{0}, {Subscript[a, 1] - Subscript[m, 1], \[Ellipsis], Subscript[a, n] - Subscript[m, n], Subscript[b, n + 1], \[Ellipsis], Subscript[b, q]}}, z] == (Product[Gamma[1 - Subscript[a, j]], {j, n + 1, p}]/ (Product[(Subscript[m, j] - 1)!, {j, 1, n}] Product[Gamma[1 - Subscript[b, j]], {j, n + 1, q}])) Sum[\[Ellipsis] Sum[(1/(1 - Subscript[a, k] + Subscript[j, k])) Product[(Pochhammer[1 - Subscript[m, l], Subscript[j, l]]/ Subscript[j, l]!) Product[If[i == k, 1, 1/(Subscript[a, k] + Subscript[j, i] - Subscript[a, i] - Subscript[j, k])] HypergeometricPFQ[{1 - Subscript[a, k] + Subscript[j, k], 1 - Subscript[a, n + 1], \[Ellipsis], 1 - Subscript[a, p]}, {2 - Subscript[a, k] + Subscript[j, k], 1 - Subscript[b, n + 1], \[Ellipsis], 1 - Subscript[b, q]}, -z], {i, 1, n}], {l, 1, n}], {Subscript[j, n], 0, Subscript[m, n] - 1}], {k, 1, n}, {Subscript[j, 1], 0, Subscript[m, 1] - 1}] /; Element[Subscript[m, n], Integers] && Subscript[m, n] > 0 && n <= q

 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", "p"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["m", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "n"], "-", SubscriptBox["m", "n"]]], ",", SubscriptBox["b", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "p"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["a", "j"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["m", "j"], "-", "1"]], ")"]], "!"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "q"], RowBox[List["Gamma", "[", RowBox[List["1", "-", SubscriptBox["b", "j"]]], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], RowBox[List[SubscriptBox["m", "1"], "-", "1"]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], RowBox[List[SubscriptBox["m", "n"], "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["m", "l"]]], ",", SubscriptBox["j", "l"]]], "]"]], RowBox[List[SubscriptBox["j", "l"], "!"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], "n"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["i", "\[Equal]", "k"]], ",", "1", ",", FractionBox["1", RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "i"], "-", SubscriptBox["a", "i"], "-", SubscriptBox["j", "k"]]]]]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], ",", RowBox[List["1", "-", SubscriptBox["a", RowBox[List["n", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], ",", RowBox[List["1", "-", SubscriptBox["b", RowBox[List["n", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["m", "n"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["m", "n"], ">", "0"]], "\[And]", RowBox[List["n", "\[LessEqual]", "q"]]]]]]]]

 MathML Form

 G p , q + 1 1 , p ( z a 1 , , a p 0 , a 1 - m 1 , , a n - m n , b n + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", RowBox[List["q", "+", "1"]]]], RowBox[List["1", ",", "p"]]], "\[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", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["m", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "n"], "-", SubscriptBox["m", "n"]]], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] j = n + 1 p Γ ( 1 - a j ) ( j = 1 n ( m j - 1 ) ! ) j = n + 1 q Γ ( 1 - b j ) k = 1 n j 1 = 0 m 1 - 1 j n = 0 m n - 1 1 j k - a k + 1 l = 1 n ( 1 - m l ) j l TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "l"]]], ")"]], SubscriptBox["j", "l"]], Pochhammer] j l ! j = 1 j k n 1 a k - a i + j i - j k p - n + 1 F q - n + 1 ( j k - a k + 1 , 1 - a n + 1 , , 1 - a p ; j k - a k + 2 , 1 - b n + 1 , , 1 - b q ; - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "-", "n", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["q", "-", "n", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["j", "k"], "-", SubscriptBox["a", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["n", "+", "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[SubscriptBox["j", "k"], "-", SubscriptBox["a", "k"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["n", "+", "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["-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; m n + n q G p , q + 1 1 , p ( z a 1 , , a p 0 , a 1 - m 1 , , a n - m n , b n + 1 , , b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", RowBox[List["q", "+", "1"]]]], RowBox[List["1", ",", "p"]]], "\[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", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["m", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "n"], "-", SubscriptBox["m", "n"]]], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] j = n + 1 p Γ ( 1 - a j ) ( j = 1 n ( m j - 1 ) ! ) j = n + 1 q Γ ( 1 - b j ) k = 1 n j 1 = 0 m 1 - 1 j n = 0 m n - 1 1 j k - a k + 1 l = 1 n ( 1 - m l ) j l TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "l"]]], ")"]], SubscriptBox["j", "l"]], Pochhammer] j l ! j = 1 j k n 1 a k - a i + j i - j k p - n + 1 F q - n + 1 ( j k - a k + 1 , 1 - a n + 1 , , 1 - a p ; j k - a k + 2 , 1 - b n + 1 , , 1 - b q ; - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "-", "n", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["q", "-", "n", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["j", "k"], "-", SubscriptBox["a", "k"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["n", "+", "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[SubscriptBox["j", "k"], "-", SubscriptBox["a", "k"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["n", "+", "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["-", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] /; m n + n q [/itex]

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

 2001-10-29