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Cyclic identities involving Jacobi elliptic functions. (English) Zbl 1060.33026
Summary: We state and discuss numerous new mathematical identities involving Jacobi elliptic functions sn($x,m$), cn($x,m$), and dn($x,m$), where $m$ is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either $2K\left(m\right)/p$ or $4K\left(m\right)/p$, where p is an integer and $K\left(m\right)$ is the complete elliptic integral of the first kind. Each $p$-point identity of rank $r$ involves a cyclic homogeneous polynomial of degree $r$ (in Jacobi elliptic functions with $p$ equally spaced arguments) related to other cyclic homogeneous polynomials of degree $r-2$ or smaller. We algebraically demonstrate the derivation of several of our identities for specific small values of $p$ and $r$ by using standard properties of Jacobi elliptic functions. Identities corresponding to higher values of $p$ and $r$ are verified numerically using advanced mathematical software packages.
##### MSC:
 3.3e+06 Elliptic functions and integrals