Euler polynomials and Stieltjes-Rogers continued fractions

It is well-known that the Euler polynomials E-2n(x) with n greater than or equal to 0 can be expressed as a polynomial H-n(x(x - 1)) of x(x - 1). We e xtend H-n(u) to formal power series for n < 0 and prove several properties of the coefficients appearing in these polynomials or series, which general ize some recent results, independently obtained by Hammersley [7] and Horad am [8], and answer a question of Kreweras [9]. We also deduce several conti nued fraction expansions for the generating function of Euler polynomials, some of these formulae had been published by Stieltjes [14] and by Rogers [ 12] without proof. These formulae generalize our earlier results concerning Genocchi numbers, Euler numbers and Springer numbers [5, 4].