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 JacobiDN

 http://functions.wolfram.com/09.29.16.0071.01

 Input Form

 Sum[Product[JacobiDN[z + (2 (k + Subscript[n, l]) EllipticK[m])/p, m], {l, 0, r - 1}], {k, 0, p - 1}]/Sum[JacobiDN[z + (2 k EllipticK[m])/p, m], {k, 0, p - 1}] == Sum[Product[JacobiDN[(2 (k + Subscript[n, l]) EllipticK[m])/p, m], {l, 0, r - 1}], {k, 0, p - 1}]/Sum[JacobiDN[(2 k EllipticK[m])/p, m], {k, 0, p - 1}] /; Element[p, Integers] && p >= 2 && Element[(r - 1)/2, Integers] && r >= 3 && Subscript[n, 0] == 0 && Element[Subscript[n, l], Integers] && Inequality[1, LessEqual, Subscript[n, l], Less, p] && Subscript[n, l] < Subscript[n, l + 1]

 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[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n", "l"]]], ")"]], 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[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n", "l"]]], ")"]], 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[FractionBox[RowBox[List["r", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["r", "\[GreaterEqual]", "3"]], "\[And]", RowBox[List[SubscriptBox["n", "0"], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["n", "l"], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", SubscriptBox["n", "l"], "<", "p"]], "\[And]", RowBox[List[SubscriptBox["n", "l"], "<", SubscriptBox["n", RowBox[List["l", "+", "1"]]]]]]]]]]]

 MathML Form

 k = 0 p - 1 l = 0 r - 1 dn ( z + 2 ( k + n l ) K ( m ) p m ) k = 0 p - 1 dn ( z + 2 k K ( m ) p m ) k = 0 p - 1 l = 0 r - 1 dn ( 2 ( k + n l ) K ( m ) p m ) k = 0 p - 1 dn ( 2 k K ( m ) p m ) /; p - 2 r - 1 2 + n 0 0 n l 1 n l < p n l < n l + 1 Condition k 0 p -1 l 0 r -1 JacobiDN z 2 k Subscript n l EllipticK m p -1 m k 0 p -1 JacobiDN z 2 k EllipticK m p -1 m -1 k 0 p -1 l 0 r -1 JacobiDN 2 k Subscript n l EllipticK m p -1 m k 0 p -1 JacobiDN 2 k EllipticK m p -1 m -1 p -2 r -1 2 -1 SuperPlus Subscript n 0 0 Subscript n l Inequality 1 Subscript n l p Subscript n l Subscript n l 1 [/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[UnderoverscriptBox["\[Product]", RowBox[List["l_", "=", "0"]], RowBox[List["r_", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["n_", "l_"]]], ")"]], " ", 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[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "0"]], RowBox[List["r", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", SubscriptBox["nn", "l"]]], ")"]], " ", 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[FractionBox[RowBox[List["r", "-", "1"]], "2"], "\[Element]", "Integers"]], "&&", RowBox[List["r", "\[GreaterEqual]", "3"]], "&&", RowBox[List[SubscriptBox["n", "0"], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["nn", "l"], "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", SubscriptBox["nn", "l"], "<", "p"]], "&&", RowBox[List[SubscriptBox["nn", "l"], "<", SubscriptBox["n", RowBox[List["l", "+", "1"]]]]]]]]]]]]]

 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