Derivative[n + m - 2][AiryAi][(x + y)/t^(1/3)] == t^((1/3) (n + m - 1)) Integrate[(Derivative[n][AiryAi][x/\[Tau]^(1/3)] Derivative[m][AiryAi][y/(t - \[Tau])^(1/3)])/ (\[Tau]^((n + 1)/3) (t - \[Tau])^((m + 1)/3)), {\[Tau], 0, t}] /; Element[n, Integers] && Element[m, Integers]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> Ai </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;n&quot;, &quot;-&quot;, &quot;2&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> </mrow> <mroot> <mi> t </mi> <mn> 3 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> t </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 3 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mi> t </mi> </msubsup> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> Ai </mi> <semantics> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;n&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mfrac> <mi> x </mi> <mroot> <mi> &#964; </mi> <mn> 3 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> Ai </mi> <semantics> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;m&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mfrac> <mi> y </mi> <mroot> <mrow> <mi> t </mi> <mo> - </mo> <mi> &#964; </mi> </mrow> <mn> 3 </mn> </mroot> </mfrac> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> &#964; </mi> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> - </mo> <mi> &#964; </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 3 </mn> </mfrac> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> &#964; </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> AiryAi </ci> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <times /> <apply> <plus /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </list> </apply> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 3 </cn> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <int /> <bvar> <ci> &#964; </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> t </ci> </uplimit> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> AiryAi </ci> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <ci> &#964; </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <power /> <ci> &#964; </ci> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> </list> </apply> <apply> <ci> D </ci> <apply> <ci> AiryAi </ci> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#964; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#964; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </list> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> &#964; </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#964; </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8484; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8484; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18