Note on the Jacobi elliptic functions (Q1383961)

The Jacobi elliptic functions \(x_1(t)=\text{sn } t\), \(x_2(t)=\text{cn } t\), \(x_3(t)=\text{dn } t\) constitute the solution of the autonomous system of quadratic differential equations \(x'(t)+x(t)^2=0\), with \(x(t)=(x_1(t),x_2(t),x_3(t))\), initial values \(x(0)=(0,0,1)\), and where \[ (x_1,x_2,x_3)^2=(-x_2x_3,x_3x_1,-k^2x_1x_2), \] which defines a commutative product over \({\mathbb R}^3\) [see \textit{S. Walcher}, Algebras and differential equations, Hadronic Press, Palm Harbor, FL (1991; Zbl 0791.17002 resp. Zbl 0754.17002)]. The purpose of the paper under review is to take advantage of the commutative algebra thus defined over \({\mathbb R}^3\), denoted by \(J(k)\), to give a recursive formula for the power series expansion of the Jacobi elliptic functions. To achieve this, it is first proved that \(J(k)\) is a simple algebra equipped with a distinguished \({\mathbb Z}_2\)-grading, and that the coefficients of the power series expansion belong alternatively to the even and odd parts. A discussion over arbitrary fields of characteristic zero is also carried out.