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 JacobiDN

 http://functions.wolfram.com/09.29.18.0145.01

 Input Form

 m^(n + 1) JacobiCN[z, m]^(2 n + 1) JacobiCN[z + a, m] == (-1)^n JacobiDS[a, m]^(2 n) (m JacobiCN[a, m] + JacobiDS[a, m] (JacobiZeta[JacobiAmplitude[z + a, m], m] - JacobiZeta[JacobiAmplitude[z, m], m] - JacobiZeta[ JacobiAmplitude[a, m], m])) + (JacobiCS[a, m] JacobiNS[a, m] JacobiCN[z, m] - JacobiDS[a, m] JacobiSN[z, m] JacobiDN[z, m]) Sum[(-1)^k m^(n - k - 1) JacobiDS[a, m]^(2 k) JacobiCN[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["m", RowBox[List["n", "+", "1"]]], SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List[RowBox[List["2", "n"]], "+", "1"]]], RowBox[List["JacobiCN", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "n"]]], RowBox[List["(", RowBox[List[RowBox[List["m", " ", RowBox[List["JacobiCN", "[", RowBox[List["a", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List[RowBox[List["z", "+", "a"]], ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]], "-", RowBox[List["JacobiZeta", "[", RowBox[List[RowBox[List["JacobiAmplitude", "[", RowBox[List["a", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["JacobiCS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiNS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["JacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List["JacobiDN", "[", RowBox[List["z", ",", "m"]], "]"]]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["m", RowBox[List["n", "-", "k", "-", "1"]]], SuperscriptBox[RowBox[List["JacobiDS", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["2", "k"]]], SuperscriptBox[RowBox[List["JacobiCN", "[", RowBox[List["z", ",", "m"]], "]"]], RowBox[List[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

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

 Rule Form

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