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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==3
Case {m,n,p,q}={3,1,3,3}
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http://functions.wolfram.com/07.34.03.1003.01
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MeijerG[{{a}, {c, b - 1}}, {{b, -1 - b + 2 c, (a + b)/2 - 1}, {}}, z] ==
((2 Gamma[(b - a)/2 + 1] Gamma[-a - b + 2 c])/Gamma[1 - a + c])
z^((a - b)/2 + c - 1) (1 + z)^((a + b)/2 - c)
Hypergeometric2F1[-((a + b)/2) + c, b - c, 1 - a + c, -(1/z)] /;
!IntervalMemberQ[Interval[{-Infinity, 0}], 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> <mrow> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 1 </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> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <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["3", ",", "3"]], RowBox[List["3", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["c", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", "b", "-", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["a", "+", "b"]], "2"], "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> c </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> c </mi> <mo> - </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </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["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["c", "-", FractionBox[RowBox[List["a", "+", "b"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", "a", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </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> 3 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 1 </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> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <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["3", ",", "3"]], RowBox[List["3", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["c", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "c"]], "-", "b", "-", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["a", "+", "b"]], "2"], "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> c </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> c </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> c </mi> <mo> - </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </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["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["c", "-", FractionBox[RowBox[List["a", "+", "b"]], "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "c"]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", "a", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", "z"]]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </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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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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