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KleinInvariantJ




Mathematica Notation

Traditional Notation









Elliptic Functions > KleinInvariantJ[z] > Series representations > Exponential Fourier series





http://functions.wolfram.com/09.50.06.0002.01









  


  








Input Form





KleinInvariantJ[z] == (1/1728) (E^(-2 I Pi z) + 744 + Sum[Subscript[a, k] E^(2 k I Pi z), {k, 1, Infinity}]) /; (Subscript[a, 1] == 196884 && Subscript[a, 2] == 21493760 && Subscript[a, 3] = 864299970 && Subscript[a, 4] == 20245856256 && Subscript[a, 5] == 333202640600 && (Subscript[a, \[Nu]] = Subscript[a, 2 n + 1] + (1/2) (Subscript[a, n]^2 - Subscript[a, n]) + Sum[Subscript[a, k] Subscript[a, 2 n - k], {k, 1, n - 1}] /; n = \[Nu]/4 && Mod[\[Nu], 4] == 0) && (Subscript[a, \[Nu]] = Subscript[a, 2 n + 3] - Subscript[a, 2] Subscript[a, 2 n] + (1/2) (Subscript[a, n + 1]^2 - Subscript[a, n + 1]) + (1/2) (Subscript[a, 2 n]^2 + Subscript[a, 2 n]) + Sum[Subscript[a, k] Subscript[a, 2 n - k + 2], {k, 1, n}] - Sum[(-1)^(k - 1) Subscript[a, k] Subscript[a, 4 n - k], {k, 1, 2 n - 1}] + Sum[Subscript[a, k] Subscript[a, 4 n - 4 k], {k, 1, n - 1}] /; n = (\[Nu] - 1)/4 && Mod[\[Nu], 4] == 1) && (Subscript[a, \[Nu]] = Subscript[a, 2 n + 2] + Sum[Subscript[a, k] Subscript[a, 2 n - k + 1], {k, 1, n}] /; n = (\[Nu] - 2)/4 && Mod[\[Nu], 4] == 2) && (Subscript[a, \[Nu]] = Subscript[a, 2 n + 4] - Subscript[a, 2] Subscript[a, 2 n + 1] - (1/2) (Subscript[a, 2 n + 1]^2 - Subscript[a, 2 n + 1]) + Sum[Subscript[a, k] Subscript[a, 2 n - k + 3], {k, 1, n + 1}] - Sum[(-1)^(k - 1) Subscript[a, k] Subscript[a, 4 n - k + 2], {k, 1, 2 n}] + Sum[Subscript[a, k] Subscript[a, 4 n - 4 k + 2], {k, 1, n}] /; n = (\[Nu] - 3)/4 && Mod[\[Nu], 4] == 3))








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mi> J </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;J&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 1728 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> + </mo> <mn> 744 </mn> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mn> 196884 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mn> 21493760 </mn> </mrow> <mo> &#8743; </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> = </mo> <mrow> <mn> 864299970 </mn> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> &#10869; </mo> <mn> 20245856256 </mn> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 5 </mn> </msub> <mo> &#10869; </mo> <mn> 333202640600 </mn> </mrow> <mo> &#8743; </mo> <mtext> &#62371; </mtext> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mi> n </mi> <mn> 2 </mn> </msubsup> <mo> - </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mtext> &#62371; </mtext> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </msub> <mo> - </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </msubsup> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mtext> &#62371; </mtext> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> </msub> <mo> - </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 3 </mn> </mrow> </msub> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 3 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <semantics> <mrow> <mi> J </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;J&quot;, &quot;(&quot;, TagBox[&quot;z&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[KleinInvariantJ[Slot[1]]]]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 1728 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> + </mo> <mn> 744 </mn> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mn> 196884 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mn> 21493760 </mn> </mrow> <mo> &#8743; </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> </mrow> <mo> = </mo> <mrow> <mn> 864299970 </mn> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 4 </mn> </msub> <mo> &#10869; </mo> <mn> 20245856256 </mn> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 5 </mn> </msub> <mo> &#10869; </mo> <mn> 333202640600 </mn> </mrow> <mo> &#8743; </mo> <mtext> &#62371; </mtext> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mi> n </mi> <mn> 2 </mn> </msubsup> <mo> - </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mtext> &#62371; </mtext> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> </msub> <mo> - </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </msubsup> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mtext> &#62371; </mtext> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 2 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> &#957; </mi> </msub> <mo> = </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 4 </mn> </mrow> </msub> <mo> - </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </msubsup> <mo> - </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 3 </mn> </mrow> </msub> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mi> k </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <msub> <mi> a </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> </mrow> <mo> /; </mo> <mi> n </mi> </mrow> <mo> = </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 4 </mn> </mrow> <mo> &#10869; </mo> <mn> 3 </mn> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29