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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] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself > Representation of fundamental system solutions near point z==ComplexInfinity for p>=q in the general case





http://functions.wolfram.com/07.34.13.0003.01









  


  








Input Form





(-1)^(m + n - p) z Fold[Function[{f, l}, z D[f, z] + (1 - Subscript[a, l]) f], w[z], {1, \[Ellipsis], p}] - Fold[Function[{f, k}, z D[f, z] - Subscript[b, k] f], w[z], {1, \[Ellipsis], q}] == 0 /; w[z] == Subscript[c, 1] 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] + Sum[Subscript[c, k] MeijerG[{{1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, q]}, {}}, {{1 - Subscript[a, k]}, {1 - Subscript[a, 1], \[Ellipsis], 1 - Subscript[a, k - 1], 1 - Subscript[a, k + 1], \[Ellipsis], 1 - Subscript[a, p]}}, (-1)^(q - m - n + 1)/z], {k, 2, p}] && p >= q && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= p && 1 <= k <= p, !Element[Subscript[a, j] - Subscript[a, k], Integers]]








Standard Form





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MathML Form







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</mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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> <mn> 1 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </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;m&quot;, &quot;,&quot;, &quot;n&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[&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;n&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, RowBox[List[&quot;n&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;a&quot;, &quot;p&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[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;, &quot;m&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;b&quot;, RowBox[List[&quot;m&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> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mi> q </mi> <mo> , </mo> <mi> p </mi> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> q </mi> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> q </mi> </mrow> </msup> <mi> z </mi> </mfrac> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;q&quot;, &quot;,&quot;, &quot;p&quot;]], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;q&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;m&quot;, &quot;-&quot;, &quot;n&quot;, &quot;+&quot;, &quot;q&quot;]]], &quot;z&quot;], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;k&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, 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[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, RowBox[List[&quot;k&quot;, &quot;-&quot;, &quot;1&quot;]]]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&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[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;p&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8805; </mo> <mi> q </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mo> &#8704; </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> j </mi> <mo> &#8804; </mo> <mi> p </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> k </mi> <mo> &#8804; </mo> <mi> p </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m_", "+", "n_", "-", "p_"]]], " ", "z_", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "l_"]], "}"]], ",", RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "f_"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a_", "l_"]]], ")"]], " ", "f_"]]]]]], "]"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]_", ",", "p_"]], "}"]]]], "]"]]]], "-", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f_", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z_", " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], "f_"]]]], "-", RowBox[List[SubscriptBox["b_", "k"], " ", "f_"]]]]]], "]"]], ",", RowBox[List["w", "[", "z_", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]_", ",", "q_"]], "}"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", 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"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "p"], RowBox[List[SubscriptBox["c", "k"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], ",", RowBox[List["1", "-", SubscriptBox["a", RowBox[List["k", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "p"]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n", "+", "1"]]], "z"]]], "]"]]]]]]]]]], "&&", RowBox[List["p", "\[GreaterEqual]", "q"]], "&&", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "&&", RowBox[List["j", "\[NotEqual]", "k"]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "p"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "p"]]]]]]], RowBox[List["(", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", "k"]]], "\[Element]", "Integers"]]]], ")"]]]]]]]]]]]]


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





2001-10-29