Hermite polynomials and their applications associated with bernoulli and euler numbers

Abstract

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. Let Pn = { p(x) ∈ ℚ[ x ] | deg p(x) ≤ n } be the (n + 1) -dimensional vector space over ℚ. Then we show that { H0 (x), H1(x), ⋯, Hn (x) } is a good basis for the space P n for our purpose of arithmetical and combinatorial applications. © 2012 Dae San Kim et al.