Sum[((-1)^k Binomial[2 m, m - k] HermiteH[n - k, Sqrt[2] z] HermiteH[n + k, Sqrt[2] z])/2^n, {k, -m, m}] == (((2 m)! (n - m)!)/m!) Sum[(2^(k - n) Binomial[k - 1, m - 1] HermiteH[n - k, Sqrt[2] z]^2)/(n - k)!, {k, m, n}] /; Element[n, Integers] && n >= 1 && Element[n, Integers] && n >= 1 && n >= m

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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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mi> m </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;m&quot;]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;k&quot;]], Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! 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2002-12-18