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 JacobiDN

 http://functions.wolfram.com/09.29.16.0057.01

 Input Form

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

 Standard Form

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 MathML Form

 k = 0 p - 1 dn ( z + 2 k K ( m ) p m ) 2 ( dn ( z + 2 ( k - n ) K ( m ) p m ) + dn ( z + 2 ( k + n ) K ( m ) p m ) ) k = 0 p - 1 dn ( z + 2 k K ( m ) p m ) k = 0 p - 1 dn ( 2 k K ( m ) p m ) 2 ( dn ( 2 ( k - n ) K ( m ) p m ) + dn ( 2 ( k + n ) K ( m ) p m ) ) k = 0 p - 1 dn ( 2 k K ( m ) p m ) /; p - 2 n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] 1 n p + p mod 2 FE`Conversion`Private`p 2 2 Condition k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m 2 JacobiDN z 2 k -1 n EllipticK m p -1 m JacobiDN z 2 k n EllipticK m p -1 m k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m -1 k 0 p -1 JacobiDN 2 k EllipticK m p -1 m 2 JacobiDN 2 k -1 n EllipticK m p -1 m JacobiDN 2 k n EllipticK m p -1 m k 0 p -1 JacobiDN 2 k EllipticK m p -1 m -1 p -2 n 1 n p FE`Conversion`Private`p 2 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "n_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "n_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]], ")"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "n"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]], ")"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "2"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "n", "\[LessEqual]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["Mod", "[", RowBox[List["p", ",", "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