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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 ${E}_{n}$ which satisfies the identity ${E}_{n}\left(x\right)+{E}_{n}\left(x+1\right)=2{x}^{n}$. It is known that its roots are symmetric with respect to the point $\frac{1}{2}$. 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 ${E}_{n}$ and their number. Partial results had been obtained previously by 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 ${x}_{1}^{\left(n\right)}$, ${x}_{2}^{\left(n\right)}$,..., ${x}_{N}^{\left(n\right)}$ be these roots arranged in order of increasing magnitude. Very simple arguments permit to determine for $n>5$ and $1 an interval of length $\frac{1}{2}$ which contains ${x}_{r}^{\left(n\right)}$. After that, bounds for ${x}_{N}^{\left(n\right)}$ are given. This permits to determine an integer $\nu \left(n\right)$ such that $N$ is either $\nu \left(n\right)$ or $\nu \left(n\right)+2$. As $n$ tends to infinity $\nu \left(n\right)\sim n/\pi e·$

Finally it is proved that, as $k$ tends to infinity, ${x}_{r}^{\left(2k\right)}$ tends to $r$ and ${x}_{r}^{\left(2k-1\right)}$ tends to $r-\frac{1}{2}$, and precise information is given on the difference ${x}_{r}^{\left(n\right)}-{\ell }_{n}\left(r\right)$ where ${\ell }_{n}\left(r\right)=r$ if $n$ is even and $r-\frac{1}{2}$ if $n$ is odd.

Reviewer: H. Delange
##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials
##### 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). [3] Inkeri, K.: The real roots of Bernoulli polynomials. Ann. Univ. Turk., Ser. A37, 3-19 (1959). [4] Lindelöf, E.: Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Paris: Gauthier-Villars. 1905.