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 JacobiDN

 http://functions.wolfram.com/09.29.16.0059.01

 Input Form

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

 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[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n", "1"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n", "2"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]]]]]], ")"]], "/", RowBox[List["(", 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"]], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n", "1"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n", "2"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "2"]], "\[And]", RowBox[List[SubscriptBox["n", "1"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", SubscriptBox["n", "1"], "<", SubscriptBox["n", "2"], "<", "p"]]]]]]]]

 MathML Form

 k = 0 p - 1 dn ( z + 2 k K ( m ) p m ) dn ( z + 2 K ( m ) ( k + n 1 ) p m ) dn ( z + 2 K ( m ) ( k + n 2 ) p m ) k = 0 p - 1 dn ( z + 2 k K ( m ) p m ) k = 0 p - 1 dn ( 2 k K ( m ) p m ) dn ( 2 ( k + n 1 ) K ( m ) p m ) dn ( 2 ( k + n 2 ) K ( m ) p m ) k = 0 p - 1 dn ( 2 k K ( m ) p m ) /; p - 2 n 1 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] n 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] 1 n 1 < n 2 < p Condition k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m JacobiDN z 2 EllipticK m k Subscript n 1 p -1 m JacobiDN z 2 EllipticK m k Subscript n 2 p -1 m k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m -1 k 0 p -1 JacobiDN 2 k EllipticK m p -1 m JacobiDN 2 k Subscript n 1 EllipticK m p -1 m JacobiDN 2 k Subscript n 2 EllipticK m p -1 m k 0 p -1 JacobiDN 2 k EllipticK m p -1 m -1 p -2 Subscript n 1 Subscript n 2 Inequality 1 Subscript n 1 Subscript n 2 p [/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[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n_", "1"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n_", "2"]]], ")"]], " ", 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_"]], "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["nn", "1"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["nn", "2"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "2"]], "&&", RowBox[List[SubscriptBox["nn", "1"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["nn", "2"], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", SubscriptBox["nn", "1"], "<", SubscriptBox["nn", "2"], "<", "p"]]]]]]]]]]

 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