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 JacobiDN

 http://functions.wolfram.com/09.29.16.0117.01

 Input Form

 Sum[JacobiDN[z + 2 k (EllipticK[m]/p), m]^2 JacobiDN[z + 2 (k + r) (EllipticK[m]/p), m]^2, {k, 0, p - 1}] == -2 JacobiCS[2 r (EllipticK[m]/p), m]^2 Sum[JacobiDN[z + 2 k (EllipticK[m]/p), m]^2, {k, 0, p - 1}] + (p/(2 EllipticK[m])) (Integrate[JacobiDN[t, m]^2 JacobiDN[t + 2 r (EllipticK[m]/p), m]^2, {t, 0, 2 EllipticK[m]}] + 4 EllipticE[m] JacobiCS[2 r (EllipticK[m]/p), m]^2) /; 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[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["2", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "2"], SuperscriptBox[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"]], "]"]], "2"]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["2", "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["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["2", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "2"]]]]], "+", RowBox[List[FractionBox["p", RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]]], RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["t", ",", "m"]], "]"]], "2"], SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["t", "+", RowBox[List["2", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]], "2"], RowBox[List["\[DifferentialD]", "t"]]]]]], "+", RowBox[List["4", RowBox[List["EllipticE", "[", "m", "]"]], SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["2", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"]]]]], ")"]]]]]]]], "/;", 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 dn ( z + 2 k K ( m ) p m ) 2 dn ( z + 2 ( k + r ) K ( m ) p m ) 2 p 2 K ( m ) ( 0 2 K ( m ) dn ( t m ) 2 dn ( t + 2 r K ( m ) p m ) 2 t + 4 E ( m ) cs ( 2 r K ( m ) p m ) 2 ) - 2 cs ( 2 r K ( m ) p m ) 2 k = 0 p - 1 dn ( z + 2 k K ( m ) p m ) 2 /; p + r + r < p - 1 Condition k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m 2 JacobiDN z 2 k r EllipticK m p -1 m 2 p 2 EllipticK m -1 t 0 2 EllipticK m JacobiDN t m 2 JacobiDN t 2 r EllipticK m p -1 m 2 4 EllipticE m JacobiCS 2 r EllipticK m p -1 m 2 -1 2 JacobiCS 2 r EllipticK m p -1 m 2 k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m 2 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[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "2"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], "2"]]]]], "+", FractionBox[RowBox[List["p", " ", RowBox[List["(", RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]]], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["t", ",", "m"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["t", "+", FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], "2"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "+", RowBox[List["4", " ", RowBox[List["EllipticE", "[", "m", "]"]], " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "2"]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["EllipticK", "[", "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