Generalized Meijer G-function: Identities (formula 07.35.17.0003)

MeijerG

$Mathematica Notation$

Hypergeometric Functions

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

Identities

Functional identities

http://functions.wolfram.com/07.35.17.0003.01

Input Form

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]}}, z, r] == (2 Pi)^\[Xi] r^\[Theta] MeijerG[{{Subscript[a, 1]/u, \[Ellipsis], (Subscript[a, 1] + u - 1)/u, \[Ellipsis], Subscript[a, n]/u, \[Ellipsis], (Subscript[a, n] + u - 1)/u}, {Subscript[a, n + 1]/u, \[Ellipsis], (Subscript[a, n + 1] + u - 1)/u, \[Ellipsis], Subscript[a, p]/u, \[Ellipsis], (Subscript[a, p] + u - 1)/u}}, {{Subscript[b, 1]/u, \[Ellipsis], (Subscript[b, 1] + u - 1)/u, \[Ellipsis], Subscript[b, m]/u, \[Ellipsis], (Subscript[b, m] + u - 1)/ u}, {Subscript[b, m + 1]/u, \[Ellipsis], (Subscript[b, m + 1] + u - 1)/ u, \[Ellipsis], Subscript[b, q]/u, \[Ellipsis], (Subscript[b, q] + u - 1)/u}}, u^(r (p - q)) z, 1/v] /; r == u/v && Element[u, Integers] && u > 0 && Element[v, Integers] && v > 0 && \[Xi] == ((p + q)/2 - m - n) (u - 1) && \[Theta] == Sum[Subscript[b, k], {k, 1, q}] - Sum[Subscript[a, k], {k, 1, p}] + (p - q)/2 + 1

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", "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"]]], "}"]]]], "}"]], ",", "z", ",", "r"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], "\[Xi]"], " ", SuperscriptBox["r", "\[Theta]"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", "1"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["a", "n"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "u", "-", "1"]], "u"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["a", "p"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "u", "-", "1"]], "u"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", "1"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["b", "m"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "u", "-", "1"]], "u"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["b", "q"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "u", "-", "1"]], "u"]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox["u", RowBox[List["r", RowBox[List["(", RowBox[List["p", "-", "q"]], ")"]]]]], " ", "z"]], ",", FractionBox["1", "v"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["r", "\[Equal]", FractionBox["u", "v"]]], "\[And]", RowBox[List["Element", "[", RowBox[List["u", ",", "Integers"]], "]"]], "\[And]", RowBox[List["u", ">", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["v", ",", "Integers"]], "]"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["\[Xi]", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["p", "+", "q"]], "2"], "-", "m", "-", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["u", "-", "1"]], ")"]]]]]], "\[And]", RowBox[List["\[Theta]", "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]], "+", FractionBox[RowBox[List["p", "-", "q"]], "2"], "+", "1"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <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> z </mi> <mo> , </mo> <mi> r </mi> </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[TagBox["z", MeijerG, Rule[Editable, True]], ",", TagBox["r", MeijerG, Rule[Editable, True]]]], MeijerG], "\[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> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mi> ξ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> θ </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> <mo> , </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> </mrow> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> u </mi> <mrow> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mi> v </mi> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> u </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> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> p </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> u </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> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["p", " ", "u"]], ",", RowBox[List["q", " ", "u"]]]], RowBox[List[RowBox[List["m", " ", "u"]], ",", RowBox[List["n", " ", "u"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox["u", RowBox[List["r", " ", RowBox[List["(", RowBox[List["p", "-", "q"]], ")"]]]]], " ", "z"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "v"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[SubscriptBox["a", "1"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "n"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "p"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[SubscriptBox["b", "1"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "m"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "q"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mfrac> <mi> u </mi> <mi> v </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> u </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> v </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> ξ </mi> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> θ </mi> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <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> z </mi> <mo> , </mo> <mi> r </mi> </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[TagBox["z", MeijerG, Rule[Editable, True]], ",", TagBox["r", MeijerG, Rule[Editable, True]]]], MeijerG], "\[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> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mi> ξ </mi> </msup> <mo> ⁢ </mo> <msup> <mi> r </mi> <mi> θ </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> <mo> , </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> </mrow> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> u </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> u </mi> <mrow> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mi> v </mi> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> u </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> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> p </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> u </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> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["p", " ", "u"]], ",", RowBox[List["q", " ", "u"]]]], RowBox[List[RowBox[List["m", " ", "u"]], ",", RowBox[List["n", " ", "u"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox["u", RowBox[List["r", " ", RowBox[List["(", RowBox[List["p", "-", "q"]], ")"]]]]], " ", "z"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "v"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[SubscriptBox["a", "1"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "n"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "n"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["a", "p"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[SubscriptBox["b", "1"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "m"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "m"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[SubscriptBox["b", "q"], "u"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "u", "-", "1"]], "u"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> r </mi> <mo> ⩵ </mo> <mfrac> <mi> u </mi> <mi> v </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mi> u </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> v </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> ξ </mi> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> θ </mi> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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_"]]], "}"]]]], "}"]], ",", "z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], "\[Xi]"], " ", SuperscriptBox["r", "\[Theta]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["aa", "1"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "1"], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["aa", "n"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "n"], "+", "u", "-", "1"]], "u"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["aa", "p"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["aa", "p"], "+", "u", "-", "1"]], "u"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["bb", "1"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "1"], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "m"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "m"], "+", "u", "-", "1"]], "u"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "u", "-", "1"]], "u"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "q"], "u"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "q"], "+", "u", "-", "1"]], "u"]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox["u", RowBox[List["r", " ", RowBox[List["(", RowBox[List["p", "-", "q"]], ")"]]]]], " ", "z"]], ",", FractionBox["1", "v"]]], "]"]]]], "/;", RowBox[List[RowBox[List["r", "\[Equal]", FractionBox["u", "v"]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["u", ">", "0"]], "&&", RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]], "&&", RowBox[List["\[Xi]", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["p", "+", "q"]], "2"], "-", "m", "-", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["u", "-", "1"]], ")"]]]]]], "&&", RowBox[List["\[Theta]", "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]], "+", FractionBox[RowBox[List["p", "-", "q"]], "2"], "+", "1"]]]]]]]]]]]]

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]