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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r]
Series representations
Generalized power series
Expansions at z==infinity
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http://functions.wolfram.com/07.35.06.0002.01
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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] ==
Pi^(n - 1) Sum[(Product[Gamma[1 + Subscript[b, j] - Subscript[a, k]],
{j, 1, m}]/Product[If[j == k, 1, Sin[Pi (Subscript[a, k] -
Subscript[a, j])]] Product[Gamma[Subscript[a, k] -
Subscript[b, j]], {j, m + 1, q}], {j, 1, n}])
z^((Subscript[a, k] - 1)/r) HypergeometricPFQRegularized[
{1 + Subscript[b, 1] - Subscript[a, k], \[Ellipsis],
1 + Subscript[b, q] - Subscript[a, k]},
{1 + Subscript[a, 1] - Subscript[a, k], \[Ellipsis],
1 + Subscript[a, k - 1] - Subscript[a, k], 1 + Subscript[a, k + 1] -
Subscript[a, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[a, k]},
(-1)^(q - m - n)/z^(1/r)], {k, 1, n}] /;
(p > q || (p == q && Abs[z] > 1)) && ForAll[{j, k},
Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n,
!Element[Subscript[a, j] - Subscript[a, 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> <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> <mi> π </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> + </mo> <mn> 1 </mn> </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> k </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> k </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </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> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mfrac> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> q </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> r </mi> </mrow> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["q", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["p", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", "p"]]], HypergeometricPFQRegularized, Rule[Editable, True]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n"]]], SuperscriptBox["z", RowBox[List["1", "/", "r"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> p </mi> <mo> > </mo> <mi> q </mi> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> p </mi> <mo> ⩵ </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </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> n </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - 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</mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mfrac> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </mfrac> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> q </mi> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - 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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}
