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. Not much is known about the Taylor series expansion coefficients of the Jacobian elliptic functions sn(x, k), cn(x, k), and dn(x, k). In handbooks only the first four or five terms are given. (See for instance Abramowitz and Stegun [ 1, p. 575], Hancock's book on elliptic functions [4, p. 252 and p. 486], or Gradshteyn and Ryshik [5].) Recently, however, Alois Schett gave a "combinatorial" expression of the coefficients of the Taylor series expansion of sn(x, k) and evaluated explicitly the first eight, nontrivial, terms; see [6, p. 146]. His results were extended by Dominique Dumont [3], who gave a new combinatorial interpretation of the coefficients of sn(x, k). It is often said that no recurrence formulae exist for these coefficients ([1, p. 575] or [6, p. 143]). What is meant by that is probably that no simple recurrence formulae exist. Using differential and other equations, one can deduce a number of useful recurrence relations and, with the help of the Burmann-Lagrange theorem, even a definite (although hard to handle) expression of the coefficients in question.

Calculation of the Taylor series expansion coefficients of the Jacobian elliptic function 𝑠𝑛(𝑥,𝑘)