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JacobiSN






Mathematica Notation

Traditional Notation









Elliptic Functions > JacobiSN[z,m] > Transformations > Sums over products of four Jacobi functions





http://functions.wolfram.com/09.36.16.0137.01









  


  










Input Form





Sum[(-1)^k JacobiSN[z + 2 k (EllipticK[m]/p), m]^3 (JacobiSN[z + 2 (k + r) (EllipticK[m]/p), m] + JacobiSN[z + 2 (k - r) (EllipticK[m]/p), m]), {k, 0, p - 1}] == (2/m^2) JacobiCS[2 r (EllipticK[m]/p), m] JacobiDS[2 r (EllipticK[m]/p), m] Sum[(-1)^k JacobiDN[z + 2 k (EllipticK[m]/p), m]^2, {k, 0, p - 1}] /; Element[p/2, Integers] && p >= 2 && Element[r, Integers] && r >= 1 && GCD[p, r] == 1










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "+", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]], ")"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["JacobiCS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List["JacobiDS", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "r", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"], ",", "m"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z", "+", FractionBox[RowBox[List["2", " ", "k", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "p"]]], ",", "m"]], "]"]], "2"]]]]]]], SuperscriptBox["m", "2"]], "/;", RowBox[List[RowBox[List[FractionBox["p", "2"], "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "2"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", "\[GreaterEqual]", "1"]], "&&", RowBox[List[RowBox[List["GCD", "[", RowBox[List["p", ",", "r"]], "]"]], "\[Equal]", "1"]]]]]]]]]]










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