Asymptotic Behavior and Zeros of Euler Polynomials

Abstract

Euler polynomials are named after mathematician Leonhard Euler (1707- 1783). These are the class of polynomials {En(x)} defined by 2e xt e t + 1 = X ∞ n=0 En(x) t n n! for |t| < π. where the variables x and t can be real or complex and the integer numbers En are called Euler number. In this thesis, we mainly focus to this class of polynomials and review properties of En(x). We establish an asymptotic behavior for En(x) and determine their asymptotic zeros. We briefly explain the behavior of the real and complex zeros of En(x) for sufficiently large positive integers n. We pove for 0 ≤ x < 1 the asymptotic real zeros of En(x) approximately given by: x = 2k + n 2 for some k ∈ Z.