The Euler polynomial of degree is the unique polynomial which satisfies the identity . It is known that its roots are symmetric with respect to the point . J. Brillhart proved that for all these roots are simple [J. Reine Angew. Math. 234, 45–64 (1969; Zbl 0167.35401)]. In this paper the author studies the location of the positive roots of and their number. Partial results had been obtained previously by F. T. Howard [Pac. J. Math. 64, 181–191 (1976; Zbl 0331.10005)].
Let be the number of the positive roots of and let , ,..., be these roots arranged in order of increasing magnitude. Very simple arguments permit to determine for and an interval of length which contains . After that, bounds for are given. This permits to determine an integer such that is either or . As tends to infinity
Finally it is proved that, as tends to infinity, tends to and tends to , and precise information is given on the difference where if is even and if is odd.