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On the real roots of Euler polynomials. (English) Zbl 0653.10011
The Euler polynomial of degree $n$ is the unique polynomial $E\sb n$ which satisfies the identity $E\sb n(x)+E\sb n(x+1)=2x\sp n$. It is known that its roots are symmetric with respect to the point $\tfrac12$. {\it J. Brillhart} proved that for $n\ne 5$ 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 $E\sb n$ and their number. Partial results had been obtained previously by {\it F. T. Howard} [Pac. J. Math. 64, 181--191 (1976; Zbl 0331.10005)]. Let $N$ be the number of the positive roots of $E\sb n$ and let $x\sb 1\sp{(n)}$, $x\sb 2\sp{(n)}$,..., $x\sb N\sp{(n)}$ be these roots arranged in order of increasing magnitude. Very simple arguments permit to determine for $n>5$ and $1<r\le N-2$ an interval of length $\frac12$ which contains $x\sb r\sp{(n)}$. After that, bounds for $x\sb N\sp{(n)}$ 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, $x\sb r\sp{(2k)}$ tends to $r$ and $x\sb r\sp{(2k-1)}$ tends to $r-\frac12$, and precise information is given on the difference $x\sb r\sp{(n)}-\ell\sb n(r)$ where $\ell\sb n(r)=r$ if $n$ is even and $r-\frac12$ if $n$ is odd.
Reviewer: H. Delange

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials
Full Text:
##### References:
 [1] Brillhart, J.: On the Euler and Bernoulli polynomials. J. Reine Angew. Math.234, 45-64 (1969) · Zbl 0167.35401 · doi:10.1515/crll.1969.234.45 [2] Howard, F. T.: Roots of the Euler polynomials. Pacific J. Math.64, 181-191 (1976). · Zbl 0331.10005 [3] Inkeri, K.: The real roots of Bernoulli polynomials. Ann. Univ. Turk., Ser. A37, 3-19 (1959). · Zbl 0104.01502 [4] Lindelöf, E.: Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Paris: Gauthier-Villars. 1905. · Zbl 36.0468.01