Sum[HermiteH[Sum[Subscript[m, 1, j], {j, 1, n}], Subscript[z, 1]/Sqrt[2]]* HermiteH[Sum[Subscript[m, 2, j], {j, 1, n}], Subscript[z, 2]/Sqrt[2]] * … * HermiteH[Sum[Subscript[m, n, j], {j, 1, n}], Subscript[z, n]/Sqrt[2]]* Product[Subscript[w, i, j]^Subscript[m, i, j], {i, 1, n}, {j, i + 1, n}]/ Product[Subscript[m, i, j]!, {i, 1, n}, {j, i + 1, n}], {Subscript[m, 1, 2], 0, Infinity}, {Subscript[m, 1, 3], 0, Infinity}, … {Subscript[m, 1, n], 0, Infinity}, {Subscript[m, 2, 3], 0, Infinity}, … {Subscript[m, n - 1, n], 0, Infinity}]/2^(n^2/2) == 1/Det[W] * Exp[Sum[Subscript[z, j]^2, {j, 0, n}] - Sum[Subscript[z, j]*Subscript[r, j, k] Subscript[z, k], {j, 0, n},{k, 0, n}]]/; (n ∈ Integers && n >= 1 && Subscript[m, i, i] == 0 && (Subscript[w, i, j] == Subscript[w, j, i] /; i != j) && Subscript[w, i, i] = 1 && W == {{1, Subscript[w, 1, 2], …, Subscript[w, 1, n]}, {Subscript[w, 1, 2], 1, Subscript[w, 2, 3], …, Subscript[w, 2, n]}, {Subscript[w, 1, n], …, Subscript[w, n - 1, n], 1}})

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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> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mfrac> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> i </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> n </mi> </munderover> <msubsup> <mi> w </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> <msub> <mi> m </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </msubsup> </mrow> </mrow> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> i </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> m </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> <mo> ! </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> </msub> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> m </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> </msub> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> m </mi> <mrow> <mi> n </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> </msub> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mi> n </mi> </msub> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <semantics> <mrow> <mo> &#10072; </mo> <mi> W </mi> <mo> &#10072; </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[LeftBracketingBar]&quot;, &quot;W&quot;, &quot;\[RightBracketingBar]&quot;]], List[Det]] </annotation> </semantics> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <msubsup> <mi> z </mi> <mi> j </mi> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> z </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msub> <mi> w </mi> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> &#8290; </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> m </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> w </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> <mo> &#10869; </mo> <msub> <mi> w </mi> <mrow> <mi> j </mi> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mrow> <mo> /; </mo> <mrow> <mi> i </mi> <mo> &#8800; </mo> <mi> j </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <msub> <mi> w </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mrow> </mrow> <mo> = </mo> <mrow> <mn> 1 </mn> <mo> &#8743; </mo> <mrow> <mi> W </mi> <mo> &#10869; </mo> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <msub> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <msub> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mn> 1 </mn> </mtd> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <msub> <mi> w </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> &#8942; </mo> </mtd> <mtd> <mo> &#8942; </mo> </mtd> <mtd> <mo> &#8945; </mo> </mtd> <mtd> <mo> &#8942; </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mtd> <mtd> <mo> &#8230; </mo> </mtd> <mtd> <msub> <mi> w </mi> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mtd> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> m </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> i </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> m </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> m </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HermiteH </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 1 </cn> <ci> j </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> HermiteH </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> m </ci> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> <apply> <ci> HermiteH </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> m </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <determinant /> <ci> W </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> j </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> m </ci> <ci> i </ci> <ci> i </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> i </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> j </ci> <ci> i </ci> </apply> </apply> <apply> <neq /> <ci> i </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> w </ci> <ci> i </ci> <ci> i </ci> </apply> </apply> </apply> <apply> <and /> <cn type='integer'> 1 </cn> <apply> <eq /> <ci> W </ci> <list> <list> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> w </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> w </ci> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> w </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> w </ci> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </list> <list> <ci> &#8942; </ci> <ci> &#8942; </ci> <ci> &#8945; </ci> <ci> &#8942; </ci> </list> <list> <apply> <ci> Subscript </ci> <ci> w </ci> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> w </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </list> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18