Hermite Polynomials and their Applications Associated with Bernoulli and Euler Numbers

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. LetPn={p(x)∈ℚ[x]∣deg p(x)≤n}be the(n+1)-dimensional vector space overℚ. Then we show that{H0(x),H1(x),…,Hn(x)}is a good basis for the spacePnfor our purpose of arithmetical and combinatorial applications.