Meijer G-function: Integration (formula 07.34.21.0084)

MeijerG

$Mathematica Notation$

Hypergeometric Functions

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z]

Integration

Definite integration

Integrals for classical integral transforms

http://functions.wolfram.com/07.34.21.0084.01

Input Form

Integrate[\[Tau]^(\[Alpha] - 1) (a - \[Tau])^(\[Beta] - 1) 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]}}, \[Omega] \[Tau]^(l/k)], {\[Tau], 0, a}] == ((k^\[Mu] Gamma[\[Beta]])/(l^\[Beta] ((2 Pi)^(SuperStar[c] (k - 1)) a^(1 - \[Alpha] - \[Beta])))) MeijerG[{{(1 - \[Alpha])/l, \[Ellipsis], (l - \[Alpha])/l, Subscript[a, 1]/k, \[Ellipsis], (Subscript[a, 1] + k - 1)/k, \[Ellipsis], Subscript[a, n]/k, \[Ellipsis], (Subscript[a, n] + k - 1)/ k}, {Subscript[a, n + 1]/k, \[Ellipsis], (Subscript[a, n + 1] + k - 1)/ k, \[Ellipsis], Subscript[a, p]/k, \[Ellipsis], (Subscript[a, p] + k - 1)/k}}, {{Subscript[b, 1]/k, \[Ellipsis], (Subscript[b, 1] + k - 1)/k, \[Ellipsis], Subscript[b, m]/k, \[Ellipsis], (Subscript[b, m] + k - 1)/k}, {Subscript[b, m + 1]/k, \[Ellipsis], (Subscript[b, m + 1] + k - 1)/k, \[Ellipsis], Subscript[b, q]/k, \[Ellipsis], (Subscript[b, q] + k - 1)/k, (1 - \[Alpha] - \[Beta])/l, \[Ellipsis], (l - \[Alpha] - \[Beta])/l}}, (\[Omega]^k a^l)/k^(k (q - p))] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && GCD[k, l] == 1 && (\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /; \[Sigma] == a^(-1) && s == u == v == 1 && t == Subscript[d, 1] == 0 && Subscript[c, 1] == \[Beta])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a"], RowBox[List[SuperscriptBox["\[Tau]", RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "\[Tau]"]], ")"]], RowBox[List["\[Beta]", "-", "1"]]], 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["\[Omega]", " ", SuperscriptBox["\[Tau]", FractionBox["l", "k"]]]]]], "]"]], RowBox[List["\[DifferentialD]", "\[Tau]"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["k", "\[Mu]"], SuperscriptBox["l", RowBox[List["-", "\[Beta]"]]], RowBox[List["Gamma", "[", "\[Beta]", "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], RowBox[List[SuperscriptBox["c", "*"], RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]]], SuperscriptBox["a", RowBox[List["1", "-", "\[Alpha]", "-", "\[Beta]"]]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "l"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["l", "-", "\[Alpha]"]], "l"], ",", FractionBox[SubscriptBox["a", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["a", "n"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["a", "p"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["b", "m"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["b", "q"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "-", "1"]], "k"], ",", FractionBox[RowBox[List["1", "-", "\[Alpha]", "-", "\[Beta]"]], "l"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["l", "-", "\[Alpha]", "-", "\[Beta]"]], "l"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], SuperscriptBox["a", "l"]]], SuperscriptBox["k", RowBox[List["k", RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]]]]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]], "\[And]", RowBox[List["l", "\[Element]", "Integers"]], "\[And]", RowBox[List["l", ">", "0"]], "\[And]", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "l"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List["(", RowBox[List["\[DoubleStruckCapitalC]\[DoubleStruckCapitalC]", "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Equal]", SuperscriptBox["a", RowBox[List["-", "1"]]]]], "\[And]", RowBox[List["s", "\[Equal]", "u", "\[Equal]", "v", "\[Equal]", "1"]], "\[And]", RowBox[List["t", "\[Equal]", SubscriptBox["d", "1"], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["c", "1"], "\[Equal]", "\[Beta]"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> a </mi> </msubsup> <mrow> <mrow> <msup> <mi> τ </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> β </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ω </mi> <mo> ⁢ </mo> <msup> <mi> τ </mi> <mrow> <mi> l </mi> <mo> / </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[Omega]", " ", SuperscriptBox["\[Tau]", RowBox[List["l", "/", "k"]]]]], 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", "p"], 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]] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> τ </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> k </mi> <mi> μ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> l </mi> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msup> <mi> c </mi> <mo> * </mo> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> ω </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mi> l </mi> </msup> </mrow> <msup> <mi> k </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mfrac> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mi> α </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> p </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> <mi> l </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List[RowBox[List["k", " ", "p"]], "+", "l"]], ",", RowBox[List[RowBox[List["k", " ", "q"]], "+", "l"]]]], RowBox[List[RowBox[List["k", " ", "m"]], ",", RowBox[List[RowBox[List["k", " ", "n"]], "+", "l"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], " ", SuperscriptBox["a", "l"]]], SuperscriptBox["k", RowBox[List["k", " ", RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]]]]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "l"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["l", "-", "\[Alpha]"]], "l"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "1"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "n"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "p"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[SubscriptBox["b", "1"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "m"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "q"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]", "-", "\[Beta]"]], "l"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["l", "-", "\[Alpha]", "-", "\[Beta]"]], "l"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> l </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ℂℂ </mi> <mo> /; </mo> <mrow> <mrow> <mi> σ </mi> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mi> a </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mi> u </mi> <mo> ⩵ </mo> <mi> v </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> t </mi> <mo> ⩵ </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mi> β </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> a </mi> </msubsup> <mrow> <mrow> <msup> <mi> τ </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> τ </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> β </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ω </mi> <mo> ⁢ </mo> <msup> <mi> τ </mi> <mrow> <mi> l </mi> <mo> / </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[Omega]", " ", SuperscriptBox["\[Tau]", RowBox[List["l", "/", "k"]]]]], 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", "p"], 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]] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> τ </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> k </mi> <mi> μ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> l </mi> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <msup> <mi> c </mi> <mo> * </mo> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> l </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> ω </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mi> l </mi> </msup> </mrow> <msup> <mi> k </mi> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mfrac> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mi> α </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> p </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mi> α </mi> <mo> - </mo> <mi> β </mi> </mrow> <mi> l </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List[RowBox[List["k", " ", "p"]], "+", "l"]], ",", RowBox[List[RowBox[List["k", " ", "q"]], "+", "l"]]]], RowBox[List[RowBox[List["k", " ", "m"]], ",", RowBox[List[RowBox[List["k", " ", "n"]], "+", "l"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], " ", SuperscriptBox["a", "l"]]], SuperscriptBox["k", RowBox[List["k", " ", RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]]]]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "l"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["l", "-", "\[Alpha]"]], "l"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "1"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "n"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "p"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[SubscriptBox["b", "1"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "m"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "q"], "k"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "k", "-", "1"]], "k"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]", "-", "\[Beta]"]], "l"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["l", "-", "\[Alpha]", "-", "\[Beta]"]], "l"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> l </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> gcd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ℂℂ </mi> <mo> /; </mo> <mrow> <mrow> <mi> σ </mi> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mi> a </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> s </mi> <mo> ⩵ </mo> <mi> u </mi> <mo> ⩵ </mo> <mi> v </mi> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> t </mi> <mo> ⩵ </mo> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <mi> β </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "a_"], RowBox[List[RowBox[List[SuperscriptBox["\[Tau]_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a_", "-", "\[Tau]_"]], ")"]], RowBox[List["\[Beta]_", "-", "1"]]], " ", 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["\[Omega]_", " ", SuperscriptBox["\[Tau]_", FractionBox["l_", "k"]]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "\[Tau]_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["k", "\[Mu]"], " ", SuperscriptBox["l", RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["Gamma", "[", "\[Beta]", "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "l"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["l", "-", "\[Alpha]"]], "l"], ",", FractionBox[SubscriptBox["aa", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["aa", "n"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "n"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["aa", "p"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "p"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["bb", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "m"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "m"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "q"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "q"], "+", "k", "-", "1"]], "k"], ",", FractionBox[RowBox[List["1", "-", "\[Alpha]", "-", "\[Beta]"]], "l"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["l", "-", "\[Alpha]", "-", "\[Beta]"]], "l"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], " ", SuperscriptBox["a", "l"]]], SuperscriptBox["k", RowBox[List["k", " ", RowBox[List["(", RowBox[List["q", "-", "p"]], ")"]]]]]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], RowBox[List[SuperscriptBox["c", "*"], " ", RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]]]]], " ", SuperscriptBox["a", RowBox[List["1", "-", "\[Alpha]", "-", "\[Beta]"]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]], "&&", RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", ">", "0"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["k", ",", "l"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List["(", RowBox[List["\[DoubleStruckCapitalC]\[DoubleStruckCapitalC]", "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Equal]", FractionBox["1", "a"]]], "&&", RowBox[List["s", "\[Equal]", "u", "\[Equal]", "v", "\[Equal]", "1"]], "&&", RowBox[List["t", "\[Equal]", SubscriptBox["d", "1"], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["c", "1"], "\[Equal]", "\[Beta]"]]]]]], ")"]]]]]]]]]]

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

2001-10-29

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z,r]