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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 (x)+E n (x+1)=2x n . It is known that its roots are symmetric with respect to the point 1 2. J. Brillhart proved that for n5 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 (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<rN-2 an interval of length 1 2 which contains x r (n) . After that, bounds for x N (n) are given. This permits to determine an integer ν(n) such that N is either ν(n) or ν(n)+2. As n tends to infinity ν(n)n/πe·

Finally it is proved that, as k tends to infinity, x r (2k) tends to r and x r (2k-1) tends to r-1 2, and precise information is given on the difference x r (n) - n (r) where n (r)=r if n is even and r-1 2 if n is odd.

Reviewer: H. Delange
MSC:
11B68Bernoulli 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.