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ThreeJSymbol




Mathematica Notation

Traditional Notation









Hypergeometric Functions > ThreeJSymbol[{j1,m1},{j2,m2},{j3,m3}] > Series representations > Other series representations > Series of binomial coefficients





http://functions.wolfram.com/07.39.06.0003.01









  


  








Input Form





ThreeJSymbol[{Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}] == ((-1)^(-Subscript[j, 1] + Subscript[j, 2] + Subscript[m, 3])/ Sqrt[2 Subscript[j, 3] + 1]) KroneckerDelta[Subscript[m, 1] + Subscript[m, 2], -Subscript[m, 3]] ((Sqrt[Binomial[2 Subscript[j, 1], Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3]]] Sqrt[Binomial[2 Subscript[j, 2], Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3]]])/ (Sqrt[Binomial[Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1, Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3]]] Sqrt[Binomial[2 Subscript[j, 1], Subscript[j, 1] - Subscript[m, 1]]] Sqrt[Binomial[2 Subscript[j, 2], Subscript[j, 2] - Subscript[m, 2]]] Sqrt[Binomial[2 Subscript[j, 3], Subscript[j, 3] + Subscript[m, 3]]])) Sum[(-1)^k Binomial[Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3], k] Binomial[Subscript[j, 1] - Subscript[j, 2] + Subscript[j, 3], Subscript[j, 1] - Subscript[m, 1] - k] Binomial[-Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3], Subscript[j, 2] + Subscript[m, 2] - k], {k, Max[0, Subscript[j, 2] - Subscript[j, 3] - Subscript[m, 1], Subscript[j, 1] - Subscript[j, 3] + Subscript[m, 2]], Min[Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3], Subscript[j, 1] - Subscript[m, 1], Subscript[j, 2] + Subscript[m, 2]]}] /; \[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\ \[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ][ {Subscript[j, 1], Subscript[m, 1]}, {Subscript[j, 2], Subscript[m, 2]}, {Subscript[j, 3], Subscript[m, 3]}]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ThreeJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m", "3"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["m", "3"]]]], SqrtBox[RowBox[List[RowBox[List["2", SubscriptBox["j", "3"]]], "+", "1"]]]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[SubscriptBox["m", "1"], "+", SubscriptBox["m", "2"]]], ",", RowBox[List["-", SubscriptBox["m", "3"]]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], ",", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]]]], "]"]]], " ", SqrtBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], ",", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]]]], "]"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "1"]], ",", RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]]]], "]"]]], " ", SqrtBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], ",", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["m", "1"]]]]], "]"]]], " ", SqrtBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], ",", RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["m", "2"]]]]], "]"]]], " ", SqrtBox[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "3"]]], ",", RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["m", "3"]]]]], "]"]]]]], ")"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Max", "[", RowBox[List["0", ",", RowBox[List[SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"], "-", SubscriptBox["m", "1"]]], ",", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "3"], "+", SubscriptBox["m", "2"]]]]], "]"]]]], RowBox[List["Min", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]], ",", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["m", "1"]]], ",", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"]]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Binomial", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]], ",", "k"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]], ",", RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["m", "1"], "-", "k"]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]], ",", RowBox[List[SubscriptBox["j", "2"], "+", SubscriptBox["m", "2"], "-", "k"]]]], "]"]]]]]]]]]], "/;", RowBox[List["\[ScriptCapitalP]\[ScriptH]\[ScriptY]\[ScriptS]\[ScriptI]\[ScriptC]\[ScriptA]\[ScriptL]\[ScriptCapitalQ]", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["m", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "2"], ",", SubscriptBox["m", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "3"], ",", SubscriptBox["m", "3"]]], "}"]]]], "]"]]]]]]








MathML Form







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Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29