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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] > Identities > Functional identities > General cases





http://functions.wolfram.com/07.34.17.0014.01









  


  








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] == Pi^(m + n - p) Sum[(Product[Sin[(Subscript[a, j] - Subscript[b, k]) Pi], {j, n + 1, p}]/Product[If[j == k, 1, Sin[Pi (Subscript[b, j] - Subscript[b, k])]], {j, 1, m}]) Exp[Pi I (m + n - p - 1) Subscript[b, k]] MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {}}, {{Subscript[b, k]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, k - 1], Subscript[b, k + 1], \[Ellipsis], Subscript[b, q]}}, (-1)^(p - m - n + 1) z], {k, 1, m}]








Standard Form





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MathML Form







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</mo> <mrow> <msup> <mi> &#960; </mi> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> p </mi> </munderover> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> p </mi> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;p&quot;, &quot;,&quot;, &quot;q&quot;]], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;p&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;p&quot;, &quot;-&quot;, &quot;m&quot;, &quot;-&quot;, &quot;n&quot;, &quot;+&quot;, &quot;1&quot;]]], &quot; &quot;, &quot;z&quot;]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;p&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox[&quot;b&quot;, &quot;k&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;1&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, &quot;q&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "n_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "m_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", RowBox[List["m_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List["m", "+", "n", "-", "p"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], "p"], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["b", "k"]]], ")"]], " ", "\[Pi]"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["m", "+", "n", "-", "p", "-", "1"]], ")"]], " ", SubscriptBox["b", "k"]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", SubscriptBox["b", "k"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["k", "-", "1"]]], ",", SubscriptBox["b", RowBox[List["k", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n", "+", "1"]]], " ", "z"]]]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], ")"]]]], "]"]]]], "]"]]]]]]]]]]]]]


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





2001-10-29