On the real roots of Euler polynomials
The Euler polynomial of degree \(n\) is the unique polynomial \(E_ n\) which satisfies the identity \(E_ n(x)+E_ n(x+1)=2x^ n\). It is known that its roots are symmetric with respect to the point \(\tfrac12\). \textit{J. Brillhart} proved that for \(n\neq 5\) 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 \(E_ n\) and their number. Partial results had been obtained previously by \textit{F. T. Howard} [Pac. J. Math. 64, 181--191 (1976; Zbl 0331.10005)]. Let \(N\) be the number of the positive roots of \(E_ n\) and let \(x_ 1^{(n)}\), \(x_ 2^{(n)}\),..., \(x_ N^{(n)}\) be these roots arranged in order of increasing magnitude. Very simple arguments permit to determine for \(n>5\) and \(1<r\leq N-2\) an interval of length \(\frac12\) which contains \(x_ r^{(n)}\). After that, bounds for \(x_ N^{(n)}\) are given. This permits to determine an integer \(\nu(n)\) such that \(N\) is either \(\nu(n)\) or \(\nu(n)+2\). As \(n\) tends to infinity \(\nu(n)\sim n/\pi e.\) Finally it is proved that, as \(k\) tends to infinity, \(x_ r^{(2k)}\) tends to \(r\) and \(x_ r^{(2k-1)}\) tends to \(r-\frac12\), and precise information is given on the difference \(x_ r^{(n)}-\ell_ n(r)\) where \(\ell_ n(r)=r\) if \(n\) is even and \(r-\frac12\) if \(n\) is odd.
scientific article; zbMATH DE number 4065071