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 JacobiDN

 http://functions.wolfram.com/09.29.16.0089.01

 Input Form

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

 Standard Form

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 MathML Form

 k = 0 p - 1 dn ( z + 2 K ( m ) k p m ) ( dn ( z + 2 K ( m ) ( k - r ) p m ) dn ( z + 2 K ( m ) ( k - s ) p m ) + dn ( z + 2 K ( m ) ( k + r ) p m ) dn ( z + 2 K ( m ) ( k + s ) p m ) ) - 2 ( cs ( 2 ( r - s ) K ( m ) p m ) ( cs ( 2 r K ( m ) p m ) - cs ( 2 s K ( m ) p m ) ) + cs ( 2 r K ( m ) p m ) cs ( 2 s K ( m ) p m ) ) k = 0 p - 1 dn ( z + 2 K ( m ) k p m ) /; p + r + r < p s + s < r Condition k 0 p -1 JacobiDN z 2 EllipticK m k p -1 m JacobiDN z 2 EllipticK m k -1 r p -1 m JacobiDN z 2 EllipticK m k -1 s p -1 m JacobiDN z 2 EllipticK m k r p -1 m JacobiDN z 2 EllipticK m k s p -1 m -2 JacobiCS 2 r -1 s EllipticK m p -1 m JacobiCS 2 r EllipticK m p -1 m -1 JacobiCS 2 s EllipticK m p -1 m JacobiCS 2 r EllipticK m p -1 m JacobiCS 2 s EllipticK m p -1 m k 0 p -1 JacobiDN z 2 EllipticK m k p -1 m p SuperPlus r SuperPlus r p s SuperPlus s r [/itex]

 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["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "s_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "s_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], ")"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "s", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["r", "-", "s"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "-", RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "s", " ", 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"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "1"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "r", "<", "p"]], "&&", RowBox[List["s", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "s", "<", "r"]]]]]]]]]]

 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