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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Specific values
Specialized values
Cases with m==1
Case {m,n,p,q}={1,4,4,4}
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http://functions.wolfram.com/07.34.03.0592.01
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MeijerG[{{a, c, d, e}, {}}, {{b}, {f, -1 + a, -1 + e}}, z] ==
((z^b Gamma[1 + b - c] Gamma[1 + b - d])/((1 - a + b) (1 + b - e) (-a + e)
Gamma[2 + b - f])) ((1 + b - e) (-a + f) HypergeometricPFQ[
{1 - a + b, 1 + b - c, 1 + b - d}, {2 - a + b, 2 + b - f}, -z] +
(-1 + a - b) (-e + f) HypergeometricPFQ[{1 + b - c, 1 + b - d, 1 + b - e},
{2 + b - e, 2 + b - f}, -z])
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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> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mi> a </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mi> e </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mi> f </mi> <mo> , </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["1", ",", "4"]]], "\[InvisibleApplication]", RowBox[List["(", 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⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> f </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["b", "-", "a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", 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</mo> <mrow> <mi> f </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mi> e </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> f </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", 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<apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> e </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> f </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> f </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> f </ci> </apply> <cn type='integer'> 2 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </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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