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JacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiDN[z,m] > Transformations > Sums over products of arbitrarily many Jacobi functions





http://functions.wolfram.com/09.29.16.0070.01









  


  










Input Form





Sum[Product[JacobiDN[z + (2 (k + Subscript[n, l]) EllipticK[m])/p, m], {l, 0, r - 1}], {k, 0, p - 1}] == Sum[Product[JacobiDN[(2 (k + Subscript[n, l]) EllipticK[m])/p, m], {l, 0, r - 1}], {k, 0, p - 1}] /; Element[p, Integers] && p >= 2 && Element[r/2, Integers] && r >= 2 && Subscript[n, 0] == 0 && Element[Subscript[n, l], Integers] && Inequality[1, LessEqual, Subscript[n, l], Less, p] && Subscript[n, l] < Subscript[n, l + 1]










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> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> n </mi> <mi> l </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> dn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> n </mi> <mi> l </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mi> r </mi> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> l </mi> </msub> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <msub> <mi> n </mi> <mi> l </mi> </msub> <mo> &lt; </mo> <mi> p </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> l </mi> </msub> <mo> &lt; </mo> <msub> <mi> n </mi> <mrow> <mi> l </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <product /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <product /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <ci> JacobiDN </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <plus /> <ci> p </ci> <cn type='integer'> -2 </cn> </apply> <ci> &#8469; </ci> </apply> <apply> <in /> <apply> <times /> <ci> r </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> <ci> &#8469; </ci> </apply> <apply> <ci> Inequality </ci> <cn type='integer'> 1 </cn> <leq /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> <lt /> <ci> p </ci> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> l </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <plus /> <ci> l </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l_", "=", "0"]], RowBox[List["r_", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n_", "l_"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["nn", "l"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "2"]], "&&", RowBox[List[FractionBox["r", "2"], "\[Element]", "Integers"]], "&&", RowBox[List["r", "\[GreaterEqual]", "2"]], "&&", RowBox[List[SubscriptBox["n", "0"], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["nn", "l"], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", SubscriptBox["nn", "l"], "<", "p"]], "&&", RowBox[List[SubscriptBox["nn", "l"], "<", SubscriptBox["n", RowBox[List["l", "+", "1"]]]]]]]]]]]]]










References





A. Khare, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", math-ph/0201004, (2002) http://arXiv.org/abs/math-ph/0201004

A. Khare, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", Journal of Mathematical Physics, v. 43, issue 7, pp. 3798-3806 (2002)










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





2002-03-07





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