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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r]
Representations through more general functions
Through Meijer G
Classical cases
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http://functions.wolfram.com/07.35.26.0005.01
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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, r] ==
(2 Pi)^\[Xi] r^\[Mu] MeijerG[{{Subscript[a, 1]/r, \[Ellipsis],
(Subscript[a, 1] + r - 1)/r, \[Ellipsis], Subscript[a, n]/r,
\[Ellipsis], (Subscript[a, n] + r - 1)/r}, {Subscript[a, n + 1]/r,
\[Ellipsis], (Subscript[a, n + 1] + r - 1)/r, \[Ellipsis],
Subscript[a, p]/r, \[Ellipsis], (Subscript[a, p] + r - 1)/r}},
{{Subscript[b, 1]/r, \[Ellipsis], (Subscript[b, 1] + r - 1)/r,
\[Ellipsis], Subscript[b, m]/r, \[Ellipsis], (Subscript[b, m] + r - 1)/
r}, {Subscript[b, m + 1]/r, \[Ellipsis], (Subscript[b, m + 1] + r - 1)/
r, \[Ellipsis], Subscript[b, q]/r, \[Ellipsis],
(Subscript[b, q] + r - 1)/r}}, r^(r (p - q)) z] /;
\[Xi] == ((p + q)/2 - m - n) (r - 1) &&
\[Mu] == Sum[Subscript[b, k], {k, 1, q}] - Sum[Subscript[a, k],
{k, 1, p}] + (p - q)/2 + 1 && Element[r, Integers] && r > 0
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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> <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> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> , </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mrow> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> r </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> r </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> <mtable> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> r </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> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> p </mi> </msub> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> r </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> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> r </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> r </mi> <mo> - 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</mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </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> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </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] | | |
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){kind=small}
