Cyclic Identities Involving Jacobi Elliptic Functions

We state and discuss numerous mathematical identities involving Jacobi elliptic functions sn (x,m), cn (x,m), dn (x,m), where m is the elliptic modulus parameter. In all identities, the arguments of the Jacobi functions are separated by either 2K(m)/p or 4K(m)/p, where p is an integer and K(m) 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. Identities corresponding to small values of p, r are readily established algebraically using standard properties of Jacobi elliptic functions, whereas identities with higher values of p, r are easily verified numerically using advanced mathematical software packages.