Euler Basis, Identities, and Their Applications

Let Vn={p(x)∈ℚ[x]|deg  p(x)≤n} be the (n+1)-dimensional vector space over ℚ. We show that {E0(x),E1(x),…,En(x)} is a good basis for the space Vn, for our purpose of arithmetical and combinatorial applications. Thus, if p(x)∈ℚ[x] is of degree n, then p(x)=∑l=0nblEl(x) for some uniquely determined bl∈ℚ. In this paper we develop method for computing bl from the information of p(x)