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variants of this functions
MeijerG






Mathematica Notation

Traditional Notation









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}={3,1,2,4}





http://functions.wolfram.com/07.34.03.0988.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> b </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> b </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 4 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;3&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;]]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;2&quot;]]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;2&quot;]]], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;3&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;4&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], 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<mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> MeijerG </ci> <list> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </list> </list> <list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 4 </cn> </apply> </list> </list> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> 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<apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma 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</cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 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Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29