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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Series representations
Asymptotic series expansions at z==0 for q<p
Expansions for p>q+2
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http://functions.wolfram.com/07.34.06.0026.01
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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] \[Proportional]
Sum[((Product[If[j == k, 1, Gamma[Subscript[b, j] - Subscript[b, k]]],
{j, 1, m}] Product[Gamma[1 + Subscript[b, k] - Subscript[a, j]],
{j, 1, n}])/(Product[Gamma[Subscript[a, j] - Subscript[b, k]],
{j, n + 1, p}] Product[Gamma[1 - Subscript[b, j] + Subscript[b, k]],
{j, m + 1, q}])) z^Subscript[b, k] (1 + O[z]), {k, 1, m}] +
(((2 Pi)^((1 - \[Beta])/2) Pi^(m + n - q - 1))/Sqrt[\[Beta]])
(Exp[\[Beta] E^((Pi I (q - m - n))/\[Beta]) (1/z)^(1/\[Beta])]
Sum[(Product[Sin[Pi (Subscript[a, r] - Subscript[b, j])],
{j, m + 1, q}]/Product[If[j == r, 1,
Sin[Pi (Subscript[a, r] - Subscript[a, j])]], {j, 1, n}])
E^(Pi I (q - m - n) (\[Chi] + Subscript[a, r] - 1))
(1 + O[z^(1/\[Beta])]), {r, 1, n}] +
Exp[(\[Beta] (1/z)^(1/\[Beta]))/E^((Pi I (q - m - n))/\[Beta])]
Sum[((Product[Sin[Pi (Subscript[a, r] - Subscript[b, j])],
{j, m + 1, q}]/Product[If[j == r, 1,
Sin[Pi (Subscript[a, r] - Subscript[a, j])]], {j, 1, n}])
(1 + O[z^(1/\[Beta])]))/E^(Pi I (q - m - n)
(\[Chi] + Subscript[a, r] - 1)), {r, 1, n}]) /;
(z -> 0) && p > q + 2 && \[Beta] == p - q &&
\[Chi] == (1/\[Beta]) (Sum[Subscript[b, j], {j, 1, q}] -
Sum[Subscript[a, j], {j, 1, p}] + (1 + \[Beta])/2) &&
ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= m &&
1 <= k <= m, !Element[Subscript[b, j] - Subscript[b, k], Integers]]
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<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> <mi> z </mi> <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["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", "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, True]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </munder> <mi> m </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> p </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> z </mi> <msub> <mi> b </mi> <mi> k </mi> </msub> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> β </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <msqrt> <mi> β </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> β </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> β </mi> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> β </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> r </mi> </mrow> </munder> <mi> n </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> χ </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mfrac> <mn> 1 </mn> <mi> β </mi> </mfrac> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> β </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> β </mi> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> β </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> r </mi> </mrow> </munder> <mi> n </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> χ </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mfrac> <mn> 1 </mn> <mi> β </mi> </mfrac> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> > </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> β </mi> <mo> ⩵ </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> χ </mi> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> β </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> β </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mo> ∀ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> j </mi> <mo> ≤ </mo> <mi> m </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> m </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ) </mo> </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> <mi> z </mi> <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["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", "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, True]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </munder> <mi> m </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> p </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> z </mi> <msub> <mi> b </mi> <mi> k </mi> </msub> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> β </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <msqrt> <mi> β </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> β </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> β </mi> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> β </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> r </mi> </mrow> </munder> <mi> n </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> χ </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mfrac> <mn> 1 </mn> <mi> β </mi> </mfrac> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> β </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> β </mi> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> β </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> r </mi> </mrow> </munder> <mi> n </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> χ </mi> <mo> + </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> z </mi> <mfrac> <mn> 1 </mn> <mi> β </mi> </mfrac> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> > </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> β </mi> <mo> ⩵ </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> χ </mi> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> β </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> β </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mo> ∀ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> j </mi> <mo> ≤ </mo> <mi> m </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> m </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | | |
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){kind=small}
