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SixJSymbol






Mathematica Notation

Traditional Notation









Hypergeometric Functions > SixJSymbol[{j1,j2,j3},{j4,j5,j6}] > Identities > Functional identities > Arguments changing by 1





http://functions.wolfram.com/07.40.17.0006.01









  


  










Input Form





SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[j, 4], Subscript[j, 5], Subscript[j, 6]}] == ((-Subscript[j, 3]) Sqrt[Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 2] Sqrt[-Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1] Sqrt[Subscript[j, 1] - Subscript[j, 2] + Subscript[j, 3] + 1] Sqrt[Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3]] Sqrt[Subscript[j, 3] + Subscript[j, 4] + Subscript[j, 5] + 2] Sqrt[Subscript[j, 3] - Subscript[j, 4] + Subscript[j, 5] + 1] Sqrt[Subscript[j, 3] + Subscript[j, 4] - Subscript[j, 5] + 1] Sqrt[-Subscript[j, 3] + Subscript[j, 4] + Subscript[j, 5]] SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3] + 1}, {Subscript[j, 4], Subscript[j, 5], Subscript[j, 6]}] - (Subscript[j, 3] + 1) Sqrt[Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3] + 1] Sqrt[-Subscript[j, 1] + Subscript[j, 2] + Subscript[j, 3]] Sqrt[Subscript[j, 1] - Subscript[j, 2] + Subscript[j, 3]] Sqrt[Subscript[j, 1] + Subscript[j, 2] - Subscript[j, 3] + 1] Sqrt[Subscript[j, 3] + Subscript[j, 4] + Subscript[j, 5] + 1] Sqrt[Subscript[j, 3] - Subscript[j, 4] + Subscript[j, 5]] Sqrt[Subscript[j, 3] + Subscript[j, 4] - Subscript[j, 5]] Sqrt[-Subscript[j, 3] + Subscript[j, 4] + Subscript[j, 5] + 1] SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3] - 1}, {Subscript[j, 4], Subscript[j, 5], Subscript[j, 6]}])/ ((2 Subscript[j, 3] + 1) (2 (Subscript[j, 1] (Subscript[j, 1] + 1) Subscript[j, 4] (Subscript[j, 4] + 1) + Subscript[j, 2] (Subscript[j, 2] + 1) Subscript[j, 5] (Subscript[j, 5] + 1) - Subscript[j, 3] (Subscript[j, 3] + 1) Subscript[j, 6] (Subscript[j, 6] + 1)) + (Subscript[j, 1] (Subscript[j, 1] + 1) + Subscript[j, 2] (Subscript[j, 2] + 1) - Subscript[j, 3] (Subscript[j, 3] + 1)) (Subscript[j, 3] (Subscript[j, 3] + 1) - Subscript[j, 4] (Subscript[j, 4] + 1) - Subscript[j, 5] (Subscript[j, 5] + 1))))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", SubscriptBox["j", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "4"], ",", SubscriptBox["j", "5"], ",", SubscriptBox["j", "6"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "2"]]], SqrtBox[RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "1"]]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "1"]]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"]]]], SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["j", "4"], "+", SubscriptBox["j", "5"], "+", "2"]]], SqrtBox[RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["j", "4"], "+", SubscriptBox["j", "5"], "+", "1"]]], SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["j", "4"], "-", SubscriptBox["j", "5"], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], "+", SubscriptBox["j", "4"], "+", SubscriptBox["j", "5"]]]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", RowBox[List[SubscriptBox["j", "3"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "4"], ",", SubscriptBox["j", "5"], ",", SubscriptBox["j", "6"]]], "}"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "1"]], ")"]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"], "+", "1"]]], SqrtBox[RowBox[List[RowBox[List["-", SubscriptBox["j", "1"]]], "+", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "-", SubscriptBox["j", "2"], "+", SubscriptBox["j", "3"]]]], SqrtBox[RowBox[List[SubscriptBox["j", "1"], "+", SubscriptBox["j", "2"], "-", SubscriptBox["j", "3"], "+", "1"]]], SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["j", "4"], "+", SubscriptBox["j", "5"], "+", "1"]]], SqrtBox[RowBox[List[SubscriptBox["j", "3"], "-", SubscriptBox["j", "4"], "+", SubscriptBox["j", "5"]]]], SqrtBox[RowBox[List[SubscriptBox["j", "3"], "+", SubscriptBox["j", "4"], "-", SubscriptBox["j", "5"]]]], SqrtBox[RowBox[List[RowBox[List["-", SubscriptBox["j", "3"]]], "+", SubscriptBox["j", "4"], "+", SubscriptBox["j", "5"], "+", "1"]]], RowBox[List["SixJSymbol", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["j", "1"], ",", SubscriptBox["j", "2"], ",", RowBox[List[SubscriptBox["j", "3"], "-", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["j", "4"], ",", SubscriptBox["j", "5"], ",", SubscriptBox["j", "6"]]], "}"]]]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["j", "3"]]], "+", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "1"], RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", "1"]], ")"]], SubscriptBox["j", "4"], RowBox[List["(", RowBox[List[SubscriptBox["j", "4"], "+", "1"]], ")"]]]], "+", RowBox[List[SubscriptBox["j", "2"], RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", "1"]], ")"]], SubscriptBox["j", "5"], RowBox[List["(", RowBox[List[SubscriptBox["j", "5"], "+", "1"]], ")"]]]], "-", RowBox[List[SubscriptBox["j", "3"], RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "1"]], ")"]], SubscriptBox["j", "6"], RowBox[List["(", RowBox[List[SubscriptBox["j", "6"], "+", "1"]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "1"], RowBox[List["(", RowBox[List[SubscriptBox["j", "1"], "+", "1"]], ")"]]]], "+", RowBox[List[SubscriptBox["j", "2"], RowBox[List["(", RowBox[List[SubscriptBox["j", "2"], "+", "1"]], ")"]]]], "-", RowBox[List[SubscriptBox["j", "3"], RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "1"]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "3"], RowBox[List["(", RowBox[List[SubscriptBox["j", "3"], "+", "1"]], ")"]]]], "-", RowBox[List[SubscriptBox["j", "4"], RowBox[List["(", RowBox[List[SubscriptBox["j", "4"], "+", "1"]], ")"]]]], "-", RowBox[List[SubscriptBox["j", "5"], RowBox[List["(", RowBox[List[SubscriptBox["j", "5"], "+", "1"]], ")"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







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<mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <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> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <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> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <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> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <semantics> <mo> { </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;{&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </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> <mrow> <msub> 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<mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 6 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 6 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SixJSymbol </ci> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 4 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 5 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 6 </cn> </apply> </list> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> 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Rule Form





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





2001-12-21





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