Jacobi elliptic function dn: Transformations (formula 09.29.16.0148)

JacobiDN

$Mathematica Notation$

Elliptic Functions

JacobiDN[z,m]

Transformations

Sums over products of five Jacobi functions

http://functions.wolfram.com/09.29.16.0148.01

Input Form

Sum[JacobiSN[z + 4 k (EllipticK[m]/p), m]^3 (JacobiSN[z + 4 (k + r) (EllipticK[m]/p), m]^2 + JacobiSN[z + 4 (k - r) (EllipticK[m]/p), m]^2), {k, 0, p - 1}] == (2/m) JacobiNS[4 r (EllipticK[m]/p), m]^2 Sum[JacobiSN[z + 4 k (EllipticK[m]/p), m]^3, {k, 0, p - 1}] + (2/m^2) (JacobiCS[4 r (EllipticK[m]/p), m]^2 JacobiNS[4 r (EllipticK[m]/p), m]^2 + JacobiDS[4 r (EllipticK[m]/p), m]^2 JacobiNS[4 r (EllipticK[m]/p), m]^ 2 + JacobiCS[4 r (EllipticK[m]/p), m]^2 JacobiDS[4 r (EllipticK[m]/p), m]^2 - 3 JacobiCS[4 r (EllipticK[m]/p), m] JacobiDS[4 r (EllipticK[m]/p), m] JacobiNS[4 r (EllipticK[m]/p), m]^2) Sum[JacobiSN[z + 4 k (EllipticK[m]/p), m], {k, 0, p - 1}] /; Element[p, Integers] && p >= 1 && Element[r, Integers] && Inequality[1, LessEqual, r, Less, p]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "3"], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "+", "r"]], ")"]], " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "-", "r"]], ")"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "2"]]], ")"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["2", "m"], SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "3"]]]]], "+", RowBox[List[FractionBox["2", SuperscriptBox["m", "2"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"]]], "-", RowBox[List["3", RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], RowBox[List["JacobiDS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], SuperscriptBox[RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "r", "<", "p"]]]]]]]]

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> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 2 </mn> <mi> m </mi> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <msup> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 2 </mn> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mi> cs </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> ds </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> ns </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> r </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> < </mo> <mi> p </mi> </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> <power /> <apply> <ci> JacobiSN </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> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <ci> r </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> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </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> <power /> <apply> <ci> JacobiSN </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> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> JacobiDS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> JacobiCS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiDS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> JacobiCS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> JacobiDS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <power /> <apply> <ci> JacobiNS </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </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> <cn type='integer'> 2 </cn> </apply> </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> <ci> JacobiSN </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> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <lt /> <ci> r </ci> <ci> p </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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References

A. Khare, A. Lakshminarayan, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions. II", math-ph/0207019, (2002) http://arXiv.org/abs/math-ph/0207019

A. Khare, A. Lakshminarayan, U. Sukhatme, "Cyclic Identities Involving Jacobi Elliptic Functions", Journal of Mathematical Physics, v. 44, issue 4, pp. 1822-1841 (2003)

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

2002-12-18