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http://functions.wolfram.com/09.36.16.0077.01
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Sum[JacobiDN[z + 4 k (EllipticK[m]/p), m]
(JacobiSN[z + 4 (k + r) (EllipticK[m]/p), m] +
JacobiSN[z + 4 (k - r) (EllipticK[m]/p), m]), {k, 0, p - 1}] == 0 /;
Element[p, Integers] && p >= 1 && Element[r, Integers] &&
Inequality[1, LessEqual, r, Less, p]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> dn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> sn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mi> p </mi> </mfrac> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> r </mi> <mo> < </mo> <mi> p </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> JacobiDN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <plus /> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> JacobiSN </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> k </ci> <ci> r </ci> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> r </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <lt /> <ci> r </ci> <ci> p </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p_", "-", "1"]]], RowBox[List[RowBox[List["JacobiDN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", "k", " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "+", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]], "+", RowBox[List["JacobiSN", "[", RowBox[List[RowBox[List["z_", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["k", "-", "r_"]], ")"]], " ", RowBox[List["EllipticK", "[", "m_", "]"]]]], "p_"]]], ",", "m_"]], "]"]]]], ")"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", "\[GreaterEqual]", "1"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["1", "\[LessEqual]", "r", "<", "p"]]]]]]]]]] |
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| 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) |
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Date Added to functions.wolfram.com (modification date)
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