Meijer G-function: Specific values (formula 07.34.03.0211)

MeijerG

$Mathematica Notation$

Hypergeometric Functions

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

Specific values

Specialized values

Cases with m==0

Case {m,n,p,q}={0,3,4,2}

http://functions.wolfram.com/07.34.03.0211.01

Input Form

MeijerG[{{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[a, 4]}}, {{}, {Subscript[b, 1], Subscript[b, 2]}}, z] == ((Pi^2 z^(-1 + Subscript[a, 1]) Csc[Pi (Subscript[a, 1] - Subscript[a, 2])] Csc[Pi (Subscript[a, 1] - Subscript[a, 3])])/ (Gamma[Subscript[a, 1] - Subscript[b, 1]] Gamma[Subscript[a, 1] - Subscript[b, 2]])) HypergeometricPFQRegularized[ {1 - Subscript[a, 1] + Subscript[b, 1], 1 - Subscript[a, 1] + Subscript[b, 2]}, {1 - Subscript[a, 1] + Subscript[a, 2], 1 - Subscript[a, 1] + Subscript[a, 3], 1 - Subscript[a, 1] + Subscript[a, 4]}, -(1/z)] + ((Pi^2 z^(-1 + Subscript[a, 2]) Csc[Pi (-Subscript[a, 1] + Subscript[a, 2])] Csc[Pi (Subscript[a, 2] - Subscript[a, 3])])/ (Gamma[Subscript[a, 2] - Subscript[b, 1]] Gamma[Subscript[a, 2] - Subscript[b, 2]])) HypergeometricPFQRegularized[ {1 - Subscript[a, 2] + Subscript[b, 1], 1 - Subscript[a, 2] + Subscript[b, 2]}, {1 + Subscript[a, 1] - Subscript[a, 2], 1 - Subscript[a, 2] + Subscript[a, 3], 1 - Subscript[a, 2] + Subscript[a, 4]}, -(1/z)] + ((Pi^2 z^(-1 + Subscript[a, 3]) Csc[Pi (-Subscript[a, 1] + Subscript[a, 3])] Csc[Pi (-Subscript[a, 2] + Subscript[a, 3])])/ (Gamma[Subscript[a, 3] - Subscript[b, 1]] Gamma[Subscript[a, 3] - Subscript[b, 2]])) HypergeometricPFQRegularized[ {1 - Subscript[a, 3] + Subscript[b, 1], 1 - Subscript[a, 3] + Subscript[b, 2]}, {1 + Subscript[a, 1] - Subscript[a, 3], 1 + Subscript[a, 2] - Subscript[a, 3], 1 - Subscript[a, 3] + Subscript[a, 4]}, -(1/z)] /; !IntervalMemberQ[Interval[{-Infinity, 0}], z] && !Element[Subscript[a, 2] - Subscript[a, 1], Integers] && !Element[Subscript[a, 3] - Subscript[a, 1], Integers] && !Element[Subscript[a, 3] - Subscript[a, 2], Integers]

Standard Form

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MathML Form

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</mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["a", "3"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["a", "4"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", "z"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; 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Rule Form

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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,r]