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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,r] > Identities > Functional identities





http://functions.wolfram.com/07.35.17.0003.01









  


  










Input Form





MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z, r] == (2 Pi)^\[Xi] r^\[Theta] MeijerG[{{Subscript[a, 1]/u, \[Ellipsis], (Subscript[a, 1] + u - 1)/u, \[Ellipsis], Subscript[a, n]/u, \[Ellipsis], (Subscript[a, n] + u - 1)/u}, {Subscript[a, n + 1]/u, \[Ellipsis], (Subscript[a, n + 1] + u - 1)/u, \[Ellipsis], Subscript[a, p]/u, \[Ellipsis], (Subscript[a, p] + u - 1)/u}}, {{Subscript[b, 1]/u, \[Ellipsis], (Subscript[b, 1] + u - 1)/u, \[Ellipsis], Subscript[b, m]/u, \[Ellipsis], (Subscript[b, m] + u - 1)/ u}, {Subscript[b, m + 1]/u, \[Ellipsis], (Subscript[b, m + 1] + u - 1)/ u, \[Ellipsis], Subscript[b, q]/u, \[Ellipsis], (Subscript[b, q] + u - 1)/u}}, u^(r (p - q)) z, 1/v] /; r == u/v && Element[u, Integers] && u > 0 && Element[v, Integers] && v > 0 && \[Xi] == ((p + q)/2 - m - n) (u - 1) && \[Theta] == Sum[Subscript[b, k], {k, 1, q}] - Sum[Subscript[a, k], {k, 1, p}] + (p - q)/2 + 1










Standard Form





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MathML Form







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</mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> u </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> u </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[RowBox[List[&quot;p&quot;, &quot; &quot;, &quot;u&quot;]], &quot;,&quot;, RowBox[List[&quot;q&quot;, &quot; &quot;, &quot;u&quot;]]]], RowBox[List[RowBox[List[&quot;m&quot;, &quot; &quot;, &quot;u&quot;]], &quot;,&quot;, RowBox[List[&quot;n&quot;, &quot; &quot;, &quot;u&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List[SuperscriptBox[&quot;u&quot;, RowBox[List[&quot;r&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;p&quot;, &quot;-&quot;, &quot;q&quot;]], &quot;)&quot;]]]]], &quot; &quot;, &quot;z&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;v&quot;], MeijerG, Rule[Editable, True]]]], MeijerG], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[FractionBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[SubscriptBox[&quot;a&quot;, &quot;n&quot;], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;n&quot;], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[SubscriptBox[&quot;b&quot;, &quot;m&quot;], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;m&quot;], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;u&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;+&quot;, &quot;u&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;u&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> r </mi> <mo> &#10869; </mo> <mfrac> <mi> u </mi> <mi> v </mi> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> u </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> v </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#958; </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mo> ( </mo> <mrow> <mi> u </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#952; </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29