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. Let P_n = {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), . . . , H_n(x)} is a good basis for the space P_n for our purpose of arithmetical and combinatorial applications. [ABSTRACT FROM AUTHOR]