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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==0 for p<=q in the general case





http://functions.wolfram.com/07.34.13.0005.01









  


  










Input Form





Wronskian[z^Subscript[b, 1] HypergeometricPFQ[ {1 + Subscript[b, 1] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, 1] - Subscript[a, p]}, {1 + Subscript[b, 1] - Subscript[b, 2], \[Ellipsis], 1 + Subscript[b, 1] - Subscript[b, q]}, (-1)^(p - m - n) z], \[Ellipsis], z^Subscript[b, k] HypergeometricPFQ[ {1 + Subscript[b, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[a, p]}, {1 + Subscript[b, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, k - 1], 1 + Subscript[b, k] - Subscript[b, k + 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, q]}, (-1)^(p - m - n) z], \[Ellipsis], z^Subscript[b, q] HypergeometricPFQ[{1 + Subscript[b, q] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, q] - Subscript[a, p]}, {1 + Subscript[b, q] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, q] - Subscript[b, q - 1]}, (-1)^(p - m - n) z], z] == ((-Product[Subscript[b, j] - Subscript[b, k], {k, 1, q}, {j, 1, k - 1}]) (KroneckerDelta[p, q] (1 - (-1)^(m + n - q) z)^ (-q - Sum[Subscript[b, k], {k, 1, q}] + Sum[Subscript[a, l], {l, 1, q}]) + Exp[(-(-1)^(m + n - q)) z] KroneckerDelta[p, q - 1] + UnitStep[q - p - 2]))/z^((q (q - 1))/2 - Sum[Subscript[b, k], {k, 1, q}])










Standard Form





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MathML Form







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</mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <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> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;p&quot;], SubscriptBox[&quot;F&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, RowBox[List[&quot;1&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, RowBox[List[&quot;1&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;q&quot;], &quot;-&quot;, SubscriptBox[&quot;b&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;1&quot;]]]]], &quot;,&quot;, 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;z&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <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> <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> </mrow> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> q </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </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> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> q </mi> </mrow> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> a </mi> <mi> l </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> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> q </mi> </mrow> </msup> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










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





2007-05-02





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