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SixJSymbol




Mathematica Notation

Traditional Notation









Hypergeometric Functions > SixJSymbol[{j1,j2,j3},{j4,j5,j6}] > Representations through more general functions > Through hypergeometric functions > Involving pFq





http://functions.wolfram.com/07.40.26.0002.01









  


  








Input Form





SixJSymbol[{Subscript[j, 1], Subscript[j, 2], Subscript[j, 3]}, {Subscript[j, 4], Subscript[j, 5], Subscript[j, 6]}] == (-1)^(e - 1) Pi Csc[e Pi] \[CapitalPhi][f - a, f - b, f - c, f - d, g - a, g - b, g - c, g - d] (HypergeometricPFQ[{a, b, c, d}, {e, f, g}, 1]/ (Gamma[e] Gamma[f] Gamma[g] \[CapitalPhi][1 - a, 1 - b, 1 - c, 1 - d, 1 + a - e, 1 + b - e, 1 + c - e, 1 + d - e])) /; a + b + c + d - e - f - g == -1 && \[CapitalPhi][Subscript[z, 1], Subscript[z, 2], Subscript[z, 3], Subscript[z, 4], Subscript[z, 5], Subscript[z, 6], Subscript[z, 7], Subscript[z, 8]] == Product[Sqrt[Gamma[Subscript[z, k]]], {k, 1, 8}] && a == -Subscript[j, 1] - Subscript[j, 2] + Subscript[j, 3] && b == Subscript[j, 3] - Subscript[j, 4] - Subscript[j, 5] && c == -Subscript[j, 2] - Subscript[j, 4] + Subscript[j, 6] && d == -Subscript[j, 1] - Subscript[j, 5] + Subscript[j, 6] && e == -1 - Subscript[j, 1] - Subscript[j, 2] - Subscript[j, 4] - Subscript[j, 5] && f == 1 - Subscript[j, 1] + Subscript[j, 3] - Subscript[j, 4] + Subscript[j, 6] && g == 1 - Subscript[j, 2] + Subscript[j, 3] - Subscript[j, 5] + Subscript[j, 6]








Standard Form





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MathML Form







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</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> j </mi> <mn> 4 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 5 </mn> </msub> </mtd> <mtd> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mtd> </mtr> </mtable> <mtext> &#8287; </mtext> <semantics> <mo> } </mo> <annotation encoding='Mathematica'> TagBox[StyleBox[&quot;}&quot;, Rule[SpanMaxSize, DirectedInfinity[1]]], SixJSymbol] </annotation> </semantics> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> e </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> csc </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#934; 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</mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo> , </mo> <mi> d </mi> </mrow> <mo> ; </mo> <mrow> <mi> e </mi> <mo> , </mo> <mi> f </mi> <mo> , </mo> <mi> g </mi> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;4&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;3&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;a&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;b&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;c&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;d&quot;, HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[&quot;e&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;f&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;g&quot;, HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> d </mi> <mo> - </mo> <mi> e </mi> <mo> - </mo> <mi> f </mi> <mo> - </mo> <mi> g </mi> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#934; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 3 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 4 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 5 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 6 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 7 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 8 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mn> 8 </mn> </munderover> <msqrt> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#10869; </mo> <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> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#10869; </mo> <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> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 4 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> d </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 6 </mn> </msub> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> e </mi> <mo> &#10869; </mo> <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> 4 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> f </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <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> 6 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> g </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> - </mo> <msub> <mi> j </mi> <mn> 5 </mn> </msub> <mo> + </mo> <msub> <mi> j </mi> <mn> 6 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29