Cyclic identities involving Jacobi elliptic functions. (English) Zbl 1060.33026
Summary: We state and discuss numerous new mathematical identities involving Jacobi elliptic functions sn(), cn(), and dn(), where is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either or , where p is an integer and is the complete elliptic integral of the first kind. Each -point identity of rank involves a cyclic homogeneous polynomial of degree (in Jacobi elliptic functions with equally spaced arguments) related to other cyclic homogeneous polynomials of degree or smaller. We algebraically demonstrate the derivation of several of our identities for specific small values of and by using standard properties of Jacobi elliptic functions. Identities corresponding to higher values of and are verified numerically using advanced mathematical software packages.
|33E05||Elliptic functions and integrals|