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 JacobiDN

 http://functions.wolfram.com/09.29.16.0165.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "0"]], RowBox[List["l", "-", "1"]]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", RowBox[List["2", RowBox[List["(", RowBox[List["j", "+", RowBox[List["k", " ", "r"]]]], ")"]], RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["(", RowBox[List["l", "-", "1"]], ")"]], "/", "2"]]], SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["2", "k", " ", "r", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]], ",", "m"]], "]"]], "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["l", "-", "1"]], ")"]], "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["(", RowBox[List["l", "-", "1"]], ")"]], "/", "2"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["n", "=", "1"]], "l"], RowBox[List["If", "[", RowBox[List[RowBox[List["n", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["JacobiCS", "[", RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "r", " ", RowBox[List[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", "+", RowBox[List["2", "k", " ", RowBox[List[RowBox[List["EllipticK", "[", "m", "]"]], "/", "p"]]]]]], ",", "m"]], "]"]]]]]]]], "/;", 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"]]]], "\[And]", RowBox[List[FractionBox[RowBox[List["l", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List["1", "\[LessEqual]", "l", "\[LessEqual]", "p"]]]]]]]]

 MathML Form

 j = 0 p - 1 k = 0 l - 1 dn ( z + 2 K ( m ) ( j + k r ) p m ) ( k = 1 l - 1 2 cs ( 2 k r K ( m ) p m ) 2 + 2 ( - 1 ) l - 1 2 k = 1 l - 1 2 n = 1 l If [ n k , 1 , cs ( 2 ( n - k ) r K ( m ) p m ) ] ) k = 0 p - 1 dn ( z + 2 K ( m ) k p m ) /; p + r + r < p - 1 l - 1 2 l p Condition j 0 p -1 k 0 l -1 JacobiDN z 2 EllipticK m j k r p -1 m k 1 l -1 2 -1 JacobiCS 2 k r EllipticK m p -1 m 2 2 -1 l -1 2 -1 k 1 l -1 2 -1 n 1 l If n k 1 JacobiCS 2 n -1 k r EllipticK m p -1 m k 0 p -1 JacobiDN z 2 EllipticK m k p -1 m p SuperPlus r SuperPlus r p -1 l -1 2 -1 l p [/itex]

 Rule Form

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 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