Vol. 64, No. 1, 1976

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Roots of the Euler polynomials

Frederic Timothy Howard

Vol. 64 (1976), No. 1, 181–191

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 En(x) are distributed in the closed intervaI [1, 3]. We also show how the real roots of En(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 En(x) has no roots of the form a√c-, c1, or a + b√c-, c even, or a + bi, a and b integers.

Mathematical Subject Classification
Primary: 30A08, 30A08
Secondary: 33A70
Received: 2 December 1975
Revised: 30 March 1976
Published: 1 May 1976
Frederic Timothy Howard