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 JacobiDN

 http://functions.wolfram.com/09.29.16.0060.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[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"], 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", "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]], ")"]]]]]], ")"]], "/", 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["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]]]], ")"]]]], "\[Equal]", 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"], 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", "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]], ")"]]]]]], ")"]], "/", 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["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["p", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "2"]]]]]]]]

 MathML Form

 k = 0 p - 1 ( - 1 ) k dn ( z + 2 k K ( m ) p m ) 2 ( dn ( z + 2 ( k - 1 ) K ( m ) p m ) + dn ( z + 2 ( k + 1 ) K ( m ) p m ) ) k = 0 p - 1 ( - 1 ) k dn ( z + 2 k K ( m ) p m ) k = 0 p - 1 ( - 1 ) k dn ( 2 k K ( m ) p m ) 2 ( dn ( 2 ( k - 1 ) K ( m ) p m ) + dn ( 2 ( k + 1 ) K ( m ) p m ) ) k = 0 p - 1 ( - 1 ) k dn ( 2 k K ( m ) p m ) /; p 2 + Condition k 0 p -1 -1 k JacobiDN z 2 k EllipticK m p -1 m 2 JacobiDN z 2 k -1 EllipticK m p -1 m JacobiDN z 2 k 1 EllipticK m p -1 m k 0 p -1 -1 k JacobiDN z 2 k EllipticK m p -1 m -1 k 0 p -1 -1 k JacobiDN 2 k EllipticK m p -1 m 2 JacobiDN 2 k -1 EllipticK m p -1 m JacobiDN 2 k 1 EllipticK m p -1 m k 0 p -1 -1 k JacobiDN 2 k EllipticK m p -1 m -1 p 2 -1 SuperPlus [/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["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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", "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", 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_"]], "]"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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", "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], "+", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", 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[FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["p", "2"], "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "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

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