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 JacobiDN

 http://functions.wolfram.com/09.29.16.0138.01

 Input Form

 Sum[(-1)^k JacobiDN[z + 2 k (EllipticK[m]/p), m] JacobiDN[z + 2 (k + r) (EllipticK[m]/p), m] JacobiDN[z + 2 (k + 2 r) (EllipticK[m]/p), m] JacobiDN[z + 2 (k + 3 r) (EllipticK[m]/p), m], {k, 0, p - 1}] == 2 (JacobiCS[2 r (EllipticK[m]/p), m] JacobiCS[4 r (EllipticK[m]/p), m] JacobiCS[6 r (EllipticK[m]/p), m] + JacobiCS[2 r (EllipticK[m]/p), m]^2 JacobiCS[4 r (EllipticK[m]/p), m]) Sum[(-1)^k JacobiZeta[JacobiAmplitude[z + 2 k (EllipticK[m]/p), m], m], {k, 0, p - 1}] /; Element[p/2, Integers] && p >= 2 && Element[r, Integers] && r >= 1 && GCD[p, r] == 1 && 1 - m > 0

 Standard Form

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 MathML Form

 k = 0 p - 1 ( - 1 ) k dn ( z + 2 k K ( m ) p m ) dn ( z + 2 ( k + r ) K ( m ) p m ) dn ( z + 2 ( k + 2 r ) K ( m ) p m ) dn ( z + 2 ( k + 3 r ) K ( m ) p m ) 2 ( cs ( 4 r K ( m ) p m ) cs ( 6 r K ( m ) p m ) cs ( 2 r K ( m ) p m ) + cs ( 2 r K ( m ) p m ) 2 cs ( 4 r K ( m ) p m ) ) k = 0 p - 1 ( - 1 ) k Ζ ( am ( z + 2 k K ( m ) p m ) m ) /; p 2 + r + r < p gcd ( p , r ) 1 1 - m > 0 Condition k 0 p -1 -1 k JacobiDN z 2 k EllipticK m p -1 m JacobiDN z 2 k r EllipticK m p -1 m JacobiDN z 2 k 2 r EllipticK m p -1 m JacobiDN z 2 k 3 r EllipticK m p -1 m 2 JacobiCS 4 r EllipticK m p -1 m JacobiCS 6 r EllipticK m p -1 m JacobiCS 2 r EllipticK m p -1 m JacobiCS 2 r EllipticK m p -1 m 2 JacobiCS 4 r EllipticK m p -1 m k 0 p -1 -1 k JacobiZeta JacobiAmplitude z 2 k EllipticK m p -1 m m p 2 -1 SuperPlus r SuperPlus r p p r 1 1 -1 m 0 [/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