Sum[(1/(2^k k!)) HermiteH[k, x] HermiteH[k, y], {k, 0, n}] == (1/(2^(n + 1) n! (x - y))) (HermiteH[n + 1, x] HermiteH[n, y] - HermiteH[n, x] HermiteH[n + 1, y])

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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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <mrow> <msub> <mi> H </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mn> 2 </mn> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <msub> <mi> H </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> HermiteH </ci> <ci> k </ci> <ci> x </ci> </apply> <apply> <ci> HermiteH </ci> <ci> k </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> HermiteH </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> x </ci> </apply> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> y </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> x </ci> </apply> <apply> <ci> HermiteH </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29