Euler Basis, Identities, and Their Applications

Abstract

LetVn={p(x)∈ℚ[x]|deg  p(x)≤n}be the(n+1)-dimensional vector space overℚ. We show that{E0(x),E1(x),…,En(x)}is a good basis for the spaceVn, for our purpose of arithmetical and combinatorial applications. Thus, ifp(x)∈ℚ[x]is of degreen, thenp(x)=∑l=0nblEl(x)for some uniquely determinedbl∈ℚ. In this paper we develop method for computingblfrom the information ofp(x).