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Zbl 0653.10011
Delange, Hubert
On the real roots of Euler polynomials. (English)
[J] Monatsh. Math. 106, No. 2, 115-138 (1988). ISSN 0026-9255; ISSN 1436-5081/e

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)]. \par 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 $\nu(n)\sim n/\pi e.$ \par 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.
[H. Delange]

MSC 2000:: *11B68 Bernoulli numbers, etc.

Keywords: Euler polynomial of degree n; location of the positive roots; number of the positive roots

Citations: Zbl 0167.35401; Zbl 0331.10005

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