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 JacobiCN

 http://functions.wolfram.com/09.26.16.0086.01

 Input Form

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

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

 MathML Form

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