Elliptic functions, continued fractions and doubled permutations
In this paper it is shown that the coefficients in the Taylor expansion of the elliptic function sn(z,\(\alpha)\) count the number of odd doubled permutations of 1 to \(2n+1\) having \(2k+1\) falls, and a similar result for the function cn(z,\(\alpha)\). (A doubled permutation is one such that, for each i, \(2i+1\) and \(2\sigma +2\) are of the same ordered type, of which there are four, namely `peak', `valley', `double rise' and `double fall'). The proof uses the continued fraction expansions of the Laplace transforms of sn(2,\(\alpha)\) and cn(2,\(\alpha)\), and properties of path diagrams.
scientific article; zbMATH DE number 4114634
scientific article