Polynômes d'Euler et Fractions Continues de Stieltjes-Rogers

Abstract

It is well-known that the Euler polynomials E2n(x) with n ≥ 0 can be expressed as a polynomial Hn(x(x − 1)) of x(x − 1). We extend Hn(u) to formal power series for n < 0 and prove several properties of the coefficients appearing in these polynomials or series, which generalize some recent results, independently obtained by Hammersley [7] and Horadam [8], and answer a question of Kreweras [9]. We also deduce several continued 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].