Hermite polynomials and their applications associated with Bernoulli and Euler numbers (Q714309)

Summary: We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. Let \[ {\mathbf P}_n = \{p(x) \in \mathbb Q[x] \mid \deg p(x) \leq n\} \] be the \((n+1)\)-dimensional vector space over \(\mathbb Q\). Then we show that \(\{H_0(x), H_1(x), \dots, H_n(x)\}\) is a good basis for the space \({\mathbf P}_n\) for our purpose of arithmetical and combinatorial applications.