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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==2
 
Case {m,n,p,q}={2,2,2,2}
 
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   http://functions.wolfram.com/07.34.03.0873.01
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    MeijerG[{{Subscript[a, 1], Subscript[a, 2]}, {}}, 
   {{Subscript[b, 1], Subscript[b, 2]}, {}}, z] == 
  z^(-1 + Subscript[a, 1]) Gamma[1 - Subscript[a, 1] + Subscript[b, 1]] 
   Gamma[1 - Subscript[a, 2] + Subscript[b, 1]] 
   Gamma[1 - Subscript[a, 1] + Subscript[b, 2]] 
   Gamma[1 - Subscript[a, 2] + Subscript[b, 2]] Hypergeometric2F1Regularized[
    1 - Subscript[a, 1] + Subscript[b, 1], 1 - Subscript[a, 1] + 
     Subscript[b, 2], 2 - Subscript[a, 1] - Subscript[a, 2] + 
     Subscript[b, 1] + Subscript[b, 2], 1 - 1/z] /; 
  !IntervalMemberQ[Interval[{-Infinity, 0}], z] 
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</mo>  <mrow>  <mn> 2 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ; </mo>  <mrow>  <mn> 1 </mn>  <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["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", FractionBox["1", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <mrow>  <mrow>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </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>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["2", ",", "2"]]], "\[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[SubscriptBox["a", "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <msup>  <mi> z </mi>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 1 </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>  <mn> 1 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 2 </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>  <mn> 1 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 2 </mn>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mn> 1 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  </mrow>  <mo> , </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  </mrow>  <mo> ; </mo>  <mrow>  <mn> 2 </mn>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> - </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ; </mo>  <mrow>  <mn> 1 </mn>  <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["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "1"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["1", "-", FractionBox["1", "z"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
   
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| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] |  |   |  
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