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http://functions.wolfram.com/07.37.16.0010.01
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Product[SphericalHarmonicY[Subscript[n, j], Subscript[m, j], \[CurlyTheta],
\[CurlyPhi]], {j, 1, p}] == (1/(4 Pi)^((p - 1)/2))
Product[Sqrt[2 Subscript[n, j] + 1]
Sum[\[Ellipsis] Sum[(SphericalHarmonicY[Subscript[k, p - 1],
Subscript[M, p], \[CurlyTheta], \[CurlyPhi]]/
Sqrt[2 Subscript[k, p - 1] + 1]) Product[
ClebschGordan[{Subscript[k, j - 1], 0}, {Subscript[n, j + 1], 0},
{Subscript[k, j], 0}] ClebschGordan[{Subscript[k, j - 1],
Subscript[M, j]}, {Subscript[n, j + 1], Subscript[m, j + 1]},
{Subscript[k, j], Subscript[M, j + 1]}], {j, 1, p - 1}],
{Subscript[k, p - 1], Max[Abs[Subscript[k, p - 2] - Subscript[n, p]],
Abs[Subscript[M, p]]], Subscript[k, p - 2] + Subscript[n, p]}],
{Subscript[k, 1], Max[Abs[Subscript[n, 1] - Subscript[n, 2]],
Abs[Subscript[M, 2]]], Subscript[n, 1] + Subscript[n, 2]},
{Subscript[k, 2], Max[Abs[Subscript[k, 1] - Subscript[n, 3]],
Abs[Subscript[M, 3]]], Subscript[k, 1] + Subscript[n, 3]}],
{j, 1, p}] /; Element[p, Integers] && p > 1 &&
Element[Subscript[n, k], Integers] && Subscript[n, k] >= 0 &&
Element[Subscript[m, k], Integers] && Abs[Subscript[m, k]] <=
Subscript[n, k] && Subscript[k, 0] == Subscript[n, 1] &&
Subscript[M, 0] == 0 && Subscript[M, j] == Sum[Subscript[m, k], {k, 1, j}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["n", "j"], ",", SubscriptBox["m", "j"], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["4", "\[Pi]"]], ")"]], FractionBox[RowBox[List["p", "-", "1"]], "2"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List[RowBox[List["(", SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["n", "j"]]], "+", "1"]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["n", "1"], "-", SubscriptBox["n", "2"]]], "]"]], ",", RowBox[List["Abs", "[", SubscriptBox["M", "2"], "]"]]]], "]"]]]], RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["k", "1"], "-", SubscriptBox["n", "3"]]], "]"]], ",", RowBox[List["Abs", "[", SubscriptBox["M", "3"], "]"]]]], "]"]]]], RowBox[List[SubscriptBox["k", "1"], "+", SubscriptBox["n", "3"]]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "1"]]], "=", RowBox[List["Max", "[", RowBox[List[RowBox[List["Abs", "[", RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "2"]]], "-", SubscriptBox["n", "p"]]], "]"]], ",", RowBox[List["Abs", "[", SubscriptBox["M", "p"], "]"]]]], "]"]]]], RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "2"]]], "+", SubscriptBox["n", "p"]]]], RowBox[List[FractionBox[RowBox[List["SphericalHarmonicY", "[", RowBox[List[SubscriptBox["k", RowBox[List["p", "-", "1"]]], ",", SubscriptBox["M", "p"], ",", "\[CurlyTheta]", ",", "\[CurlyPhi]"]], "]"]], SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["k", RowBox[List["p", "-", "1"]]]]], "+", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", RowBox[List["j", "+", "1"]]], ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["k", "j"], ",", "0"]], "}"]]]], "]"]], RowBox[List["ClebschGordan", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], ",", SubscriptBox["M", "j"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["n", RowBox[List["j", "+", "1"]]], ",", SubscriptBox["m", RowBox[List["j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["k", "j"], ",", SubscriptBox["M", RowBox[List["j", "+", "1"]]]]], "}"]]]], "]"]]]], ")"]]]]]]]]]]]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", ">", "1"]], "\[And]", RowBox[List[SubscriptBox["n", "k"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "k"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["m", "k"], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["m", "k"], "]"]], "\[LessEqual]", SubscriptBox["n", "k"]]], "\[And]", RowBox[List[SubscriptBox["k", "0"], "\[Equal]", SubscriptBox["n", "1"]]], "\[And]", RowBox[List[SubscriptBox["M", "0"], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["M", "j"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "j"], SubscriptBox["m", "k"]]]]]]]]]]]
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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> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mi> j </mi> </msub> <msub> <mi> m </mi> <mi> j </mi> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> M </mi> <mn> 2 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> M </mi> <mn> 3 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> </munderover> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> n </mi> <mi> p </mi> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> M </mi> <mi> p </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> n </mi> <mi> p </mi> </msub> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> M </mi> <mi> p </mi> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", "0", "\[MediumSpace]", "0"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", SubscriptBox["k", "j"], "\[MediumSpace]", "0"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> M </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", SubscriptBox["M", "j"], "\[MediumSpace]", SubscriptBox["m", RowBox[List["j", "+", "1"]]]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", SubscriptBox["k", "j"], "\[MediumSpace]", SubscriptBox["M", RowBox[List["j", "+", "1"]]]]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> m </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> m </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ≤ </mo> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> ⩵ </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> M </mi> <mn> 0 </mn> </msub> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> j </mi> </munderover> <msub> <mi> m </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> n </mi> <mi> j </mi> </msub> <msub> <mi> m </mi> <mi> j </mi> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> n </mi> <mi> j </mi> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> M </mi> <mn> 2 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> M </mi> <mn> 3 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 3 </mn> </msub> </mrow> </munderover> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow> <mi> max </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> - </mo> <msub> <mi> n </mi> <mi> p </mi> </msub> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> M </mi> <mi> p </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> n </mi> <mi> p </mi> </msub> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> Y </mi> <msub> <mi> k </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> M </mi> <mi> p </mi> </msub> </msubsup> <mo> ( </mo> <mrow> <mi> ϑ </mi> <mo> , </mo> <mi> φ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <mn> 0 </mn> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", "0", "\[MediumSpace]", "0"]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", SubscriptBox["k", "j"], "\[MediumSpace]", "0"]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mo> 〈 </mo> <mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> m </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mtext>   </mtext> <mo> ❘ </mo> <mtext>   </mtext> <mrow> <msub> <mi> k </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> n </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <mtext>   </mtext> <msub> <mi> M </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> <mo> 〉 </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftAngleBracket]", RowBox[List[RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", SubscriptBox["M", "j"], "\[MediumSpace]", SubscriptBox["m", RowBox[List["j", "+", "1"]]]]], "\[MediumSpace]", "\[VerticalSeparator]", "\[MediumSpace]", RowBox[List[SubscriptBox["k", RowBox[List["j", "-", "1"]]], "\[MediumSpace]", SubscriptBox["n", RowBox[List["j", "+", "1"]]], "\[MediumSpace]", SubscriptBox["k", "j"], "\[MediumSpace]", SubscriptBox["M", RowBox[List["j", "+", "1"]]]]]]], "\[RightAngleBracket]"]], ClebschGordan, Rule[StripWrapperBoxes, True]] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> m </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> m </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ≤ </mo> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> k </mi> <mn> 0 </mn> </msub> <mo> ⩵ </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> M </mi> <mn> 0 </mn> </msub> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> M </mi> <mi> j </mi> </msub> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> j </mi> </munderover> <msub> <mi> m </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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