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 JacobiDN

 http://functions.wolfram.com/09.29.16.0099.01

 Input Form

 Sum[JacobiCN[z + 4 k (EllipticK[m]/p), m] (JacobiCN[z + 4 (k + r) (EllipticK[m]/p), m] JacobiSN[z + 4 (k + r) (EllipticK[m]/p), m] + JacobiCN[z + 4 (k - r) (EllipticK[m]/p), m] JacobiSN[z + 4 (k - r) (EllipticK[m]/p), m]), {k, 0, p - 1}] == (2/m) JacobiNS[4 r (EllipticK[m]/p), m] (JacobiCS[4 r (EllipticK[m]/p), m] - JacobiDS[4 r (EllipticK[m]/p), m]) 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 - 1]

 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", "+", RowBox[List["4", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "+", "r"]], ")"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], 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"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["4", RowBox[List["(", RowBox[List["k", "-", "r"]], ")"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], 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"]], "]"]]]]]], ")"]]]]]], "\[Equal]", RowBox[List[FractionBox["2", "m"], " ", RowBox[List["JacobiNS", "[", RowBox[List[RowBox[List["4", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], RowBox[List["(", RowBox[List[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"]], "]"]]]], ")"]], 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", "<", RowBox[List["p", "-", "1"]]]]]]]]]]

 MathML Form

 k = 0 p - 1 cn ( z + 4 k K ( m ) p m ) ( cn ( z + 4 ( k - r ) K ( m ) p m ) sn ( z + 4 ( k - r ) K ( m ) p m ) + cn ( z + 4 ( k + r ) K ( m ) p m ) sn ( z + 4 ( k + r ) K ( m ) p m ) ) 2 m ns ( 4 r K ( m ) p m ) ( cs ( 4 r K ( m ) p m ) - ds ( 4 r K ( m ) p m ) ) k = 0 p - 1 sn ( z + 4 k K ( m ) p m ) /; p + r + r < p - 1 Condition k 0 p -1 JacobiCN z 4 k EllipticK m p -1 m JacobiCN z 4 k -1 r EllipticK m p -1 m JacobiSN z 4 k -1 r EllipticK m p -1 m JacobiCN z 4 k r EllipticK m p -1 m JacobiSN z 4 k r EllipticK m p -1 m 2 m -1 JacobiNS 4 r EllipticK m p -1 m JacobiCS 4 r EllipticK m p -1 m -1 JacobiDS 4 r EllipticK m p -1 m k 0 p -1 JacobiSN z 4 k EllipticK m p -1 m p SuperPlus r SuperPlus r p -1 [/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["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "-", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "-", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]]]], ")"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["JacobiNS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "-", RowBox[List["JacobiDS", "[", RowBox[List[FractionBox[RowBox[List["4", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]]]], "m"], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "1"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "r", "<", RowBox[List["p", "-", "1"]]]]]]]]]]]]

 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