Sum[(-1)^(Subscript[j, 2] + Subscript[j, 3] + Subscript[j, 4] + Subscript[j, 5] + Subscript[j, 6] + Subscript[j, 7] + Subscript[j, 8] + Subscript[j, 9] - Subscript[m, 2] - Subscript[m, 3] - Subscript[m, 4] - Subscript[m, 5] - Subscript[m, 6] - Subscript[m, 7] - Subscript[m, 8] - Subscript[m, 9]) ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}] ThreeJSymbol[{Subscript[j, 3], -Subscript[m, 3]}, {Subscript[j, 6], -Subscript[m, 6]}, {Subscript[j, 9], -Subscript[m, 9]}] ThreeJSymbol[{Subscript[j, 4], -Subscript[m, 4]}, {Subscript[j, 7], -Subscript[m, 7]}, {Derivative[1][Subscript[j, 1]], -Derivative[1][Subscript[m, 1]]}] ThreeJSymbol[ {Subscript[j, 5], Subscript[m, 5]}, {Subscript[j, 6], Subscript[m, 6]}, {Subscript[j, 4], Subscript[m, 4]}] ThreeJSymbol[{Subscript[j, 8], -Subscript[m, 8]}, {Subscript[j, 2], -Subscript[m, 2]}, {Subscript[j, 5], -Subscript[m, 5]}] ThreeJSymbol[{Subscript[j, 9], Subscript[m, 9]}, {Subscript[j, 7], Subscript[m, 7]}, {Subscript[j, 8], Subscript[m, 8]}], {Subscript[m, 2], -Subscript[j, 2], Subscript[j, 2]}, {Subscript[m, 3], -Subscript[j, 3], Subscript[j, 3]}, {Subscript[m, 4], -Subscript[j, 4], Subscript[j, 4]}, {Subscript[m, 5], -Subscript[j, 5], Subscript[j, 5]}, {Subscript[m, 6], -Subscript[j, 6], Subscript[j, 6]}, {Subscript[m, 7], -Subscript[j, 7], Subscript[j, 7]}, {Subscript[m, 8], -Subscript[j, 8], Subscript[j, 8]}, {Subscript[m, 9], -Subscript[j, 9], Subscript[j, 9]}] == ((-1)^(Subscript[j, 1] - Subscript[m, 1])/(2 Subscript[j, 1] + 1)) KroneckerDelta[Subscript[j, 1], Derivative[1][Subscript[j, 1]]] KroneckerDelta[Subscript[m, 1], Derivative[1][Subscript[m, 1]]] NineJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[j, 4], Subscript[j, 5], Subscript[j, 6]}, {Subscript[j, 7], Subscript[j, 8], Subscript[j, 9]}]

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"5"], ",", SubscriptBox["j", "6"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "7"], ",", SubscriptBox["j", "8"], ",", SubscriptBox["j", "9"]]], "}"]]]], "]"]]]]]]]]

<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> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 4 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 4 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 4 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 5 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 6 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 7 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 7 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 7 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 8 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> m </mi> <mn> 9 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 9 </mn> </msub> </mrow> </mrow> <msub> <mi> j </mi> <mn> 9 </mn> </msub> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 6 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 7 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 8 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 9 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 4 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 5 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 6 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 7 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 8 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 9 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 9 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 3 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 6 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 9 </mn> </msub> </mrow> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 4 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 7 </mn> </msub> </mtd> <mtd> <msubsup> <mi> j </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 4 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 7 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo> - </mo> <msubsup> <mi> m </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> </mrow> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 4 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mn> 5 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 6 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 4 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 8 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 2 </mn> </msub> </mrow> </mtd> <mtd> <mrow> <mo> - </mo> <msub> <mi> m </mi> <mn> 5 </mn> </msub> </mrow> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mo> ( </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;(&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> <mtext> &#8287; </mtext> <mtable> <mtr> <mtd> <msub> <mi> j </mi> <mn> 9 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 7 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 8 </mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> m </mi> <mn> 9 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 7 </mn> </msub> </mtd> <mtd> <msub> <mi> m </mi> <mn> 8 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> ) </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;)&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], ThreeJSymbol] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> m </mi> <mn> 1 </mn> </msub> </mrow> </msup> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msubsup> <mi> j </mi> <mn> 1 </mn> <mo> &#8242; </mo> </msubsup> </mrow> </msub> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> 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2001-12-21