Roots of the Euler polynomials

Frederic Timothy Howard

Vol. 64 (1976), No. 1, 181–191

DOI: 10.2140/pjm.1976.64.181

Abstract

In this paper we prove some new theorems about the real and complex roots of the Euler polynomials. For each n we show how the real roots of E_n(x) are distributed in the closed intervaI [1, 3]. We also show how the real roots of E_n(x) are distributed in the arbitrary interval [m,m + 1] for n sufficiently large. Finally, we prove that if a and b are nonzero rational numbers and c is a square-free integer, then E_n(x) has no roots of the form a, c≠1, or a + b, c even, or a + bi, a and b integers.

Mathematical Subject Classification

Primary: 30A08, 30A08

Secondary: 33A70

Milestones

Received: 2 December 1975

Revised: 30 March 1976

Published: 1 May 1976