Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











JacobiCN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiCN[z,m] > Transformations > Sums over products of two Jacobi functions





http://functions.wolfram.com/09.26.16.0043.01









  


  










Input Form





Sum[JacobiCN[z + (4 k EllipticK[m])/p, m] JacobiCN[z + (4 (k + n) EllipticK[m])/p, m], {k, 0, p - 1}] == Sum[JacobiCN[(4 k EllipticK[m])/p, m] JacobiCN[(4 (k + n) EllipticK[m])/p, m], {k, 0, p - 1}] /; Element[(p - 1)/2, Integers] && p >= 3 && Element[n, Integers] && 1 <= n <= (p + 1)/2










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["4", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["4", RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["p", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "3"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "n", "\[LessEqual]", FractionBox[RowBox[List["p", "+", "1"]], "2"]]]]]]]]]










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> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> <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> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </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> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> <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> <mo> &#8290; </mo> <mrow> <mi> cn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </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> <mfrac> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> n </mi> <mo> &#8804; </mo> <mfrac> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </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> <times /> <apply> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> <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> <ci> JacobiCN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <ci> n </ci> </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> <times /> <apply> <ci> JacobiCN </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> <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> <ci> JacobiCN </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <ci> n </ci> </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> <times /> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> n </ci> <apply> <times /> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <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[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "n_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiCN", "[", RowBox[List[FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox[RowBox[List["p", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "3"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "n", "\[LessEqual]", FractionBox[RowBox[List["p", "+", "1"]], "2"]]]]]]]]]]]










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





© 1998- Wolfram Research, Inc.