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Two conjectures are posed for the coefficients introduced by Alois Schett in the Taylor series expansion of the Jacobian elliptic function sn(x, k). The first conjecture is furnished with a proof revealing a procedure which might be useful when calculating further coefficients. Some of the coefficients are tabulated. The Taylor series expansion coefficients of the Jacobian elliptic function sn(x, k) were given by Schett [2], [3] for powers of x up to and including 49. In accordance with Wrigge [7] we write (1) sn(x, k) = I (_l)«-^-Í2n+1(l, k), „=o (2« + 1)! where we have the polynomials (2) s2n+xil,k)=±cr+xW. 7 = 0 The coefficients cj2n+x^ may be obtained using some rather complicated recurrence formulae or in a combinatorial way as in Schett [4] and in Dumont [1], where they are denoted by a2n+XOj. Conjecture I. The coefficient cjm) of k2J in the polynomials sm(\, k), where m = 2n + 1, is obtained as a solution to the difference equation over m, the characteristic equation of which is (3) íliu-[2ij-p) + i]Y+l = o. a p = 0 Proof. Jacobi in "Fundamenta Nova Theoriae Functionum Elhpticarum," 1829, and afterwards Glaisher [Messenger of Math., Vol. XI, 1882], [5, p. 516] gives the formula (4) d Sn"^' ^ = «(" l)sn"-2(x, k) n\l + k2)sn"ix, k) dx + «("+ l)k2snn+2ix, k). Putting n = 1, we get (5) d sn(x, k) = _ (1 + fc2)sn(jc> q + 2A:2sn3(x, k). dx Received July 7, 1980; revised December 5, 1980. 1980 Mathematics Subject Classification. Primary 33A25.

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