Identities for generalized Euler polynomials

Abstract

For $N\in \mathbb{N}$, let $T_N$ be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers $p_{\ell }^{(N)}$, defined as the coefficients in the expansion of $1/T_N(1/z)$, are provided. These coefficients give formulas for the classical Euler polynomials in terms of the so-called generalized Euler polynomials. The proofs are based on a probabilistic interpretation of the generalized Euler polynomials recently given by Klebanov et al. Asymptotics of $p_{\ell }^{(N)}$ are also provided.