2002/01/04 by Avinash Khare, Uday Sukhatme, U. Sukhatme
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Combinatorics #Degree (music) #Elliptic curve #Elliptic function #Elliptic integral #Elliptic rational functions #Homogeneous #Identity (music) #Integer (computer science) #Jacobi elliptic functions #Jacobi method #Jacobi polynomials #Mathematical analysis #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Physics #Polynomial #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quarter period #Rank (graph theory) #Theta function #math-ph #math.MP
paper · pdf · doi:10.1063/1.1484541
14 pages, 0 figures
arxiv created 2002/01/04 · openalex publication_date 2002/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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(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. 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.