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 JacobiDN

 http://functions.wolfram.com/09.29.18.0153.01

 Input Form

 JacobiDN[z, m]^(2 n) JacobiCN[z + a, m] JacobiDN[z + a, m] == (-1)^n JacobiCS[a, m]^(2 n) JacobiCN[z + a, m] JacobiDN[z + a, m] + (-1)^(n - 1) 2 n JacobiCS[a, m]^(2 n - 1) JacobiDS[a, m] JacobiNS[a, m] JacobiSN[z + a, m] - Sum[(-1)^k JacobiCS[a, m]^(2 k) (JacobiCS[a, m] JacobiDS[a, m] + 2 k JacobiDS[a, m] JacobiNC[a, m] JacobiNS[a, m]) JacobiCN[z, m] JacobiDN[z, m]^(2 (n - k) - 1), {k, 0, n - 1}] - JacobiNS[a, m] Sum[(-1)^k JacobiCS[a, m]^(2 k) (JacobiCS[a, m]^2 + 2 k JacobiDS[a, m]^2) JacobiSN[z, m] JacobiDN[z, m]^(2 (n - k - 1)), {k, 0, n - 1}] /; Element[n, Integers] && n - 1 >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["2", "n"]]], RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "n"]]], RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], "2", "n", " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", "n"]], "-", "1"]]], RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "k"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]]]], "+", RowBox[List["2", "k", " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiNC", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]]]]]], ")"]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]], "-", "1"]]]]]]], "-", RowBox[List[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "k"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "+", RowBox[List["2", "k", " ", SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]]]], ")"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["2", RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["n", "-", "1"]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 dn ( z m ) 2 n cn ( a + z m ) dn ( a + z m ) 2 ( - 1 ) n - 1 n ds ( a m ) ns ( a m ) sn ( a + z m ) cs ( a m ) 2 n - 1 + ( - 1 ) n cn ( a + z m ) dn ( a + z m ) cs ( a m ) 2 n - k = 0 n - 1 ( - 1 ) k cs ( a m ) 2 k ( cs ( a m ) ds ( a m ) + 2 k nc ( a m ) ns ( a m ) ds ( a m ) ) cn ( z m ) dn ( z m ) 2 ( n - k ) - 1 - ns ( a m ) k = 0 n - 1 ( - 1 ) k cs ( a m ) 2 k ( cs ( a m ) 2 + 2 k ds ( a m ) 2 ) sn ( z m ) dn ( z m ) 2 ( - k + n - 1 ) /; n + Condition JacobiDN z m 2 n JacobiCN a z m JacobiDN a z m 2 -1 n -1 n JacobiDS a m JacobiNS a m JacobiSN a z m JacobiCS a m 2 n -1 -1 n JacobiCN a z m JacobiDN a z m JacobiCS a m 2 n -1 k 0 n -1 -1 k JacobiCS a m 2 k JacobiCS a m JacobiDS a m 2 k JacobiNC a m JacobiNS a m JacobiDS a m JacobiCN z m JacobiDN z m 2 n -1 k -1 -1 JacobiNS a m k 0 n -1 -1 k JacobiCS a m 2 k JacobiCS a m 2 2 k JacobiDS a m 2 JacobiSN z m JacobiDN z m 2 -1 k n -1 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]], RowBox[List["2", " ", "n_"]]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", "a_"]], ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], " ", RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", "2", " ", "n", " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", " ", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]]]], "+", RowBox[List["2", " ", "k", " ", RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiNC", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]]]]]], ")"]], " ", RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]], "-", "1"]]]]]]], "-", RowBox[List[RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", " ", "k"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"], "+", RowBox[List["2", " ", "k", " ", SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], "2"]]]]], ")"]], " ", RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["n", "-", "1"]], "\[GreaterEqual]", "0"]]]]]]]]]]

 References

 A. Khare, A. Lakshminarayan, U. Sukhatme, "Local Identities Involving Jacobi Elliptic Functions", math-ph/0306028, (2003) http://arXiv.org/abs/math-ph/0306028

 Date Added to functions.wolfram.com (modification date)

 2003-08-21