Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function $\textrm {sn}(x, k)$

Abstract:

The Taylor series expansion coefficients of the Jacobian elliptic function ${\text {sn}}(x,k)$ and its power ${\text {sn}}^2(x,k)$ are studied. Recurrence formulae are given, and tables of the coefficients constructed. Using Lagrange’s inversion formula, these coefficients can be expressed in terms of Legendre polynomials.