Generalized Taylor expansions Euler-Maclaurin summation formula

Abstract

Let K be a commutative field of characteristic 0, and K[X] the algebra of polynomials in one indeterminate over K (Alg., IV. 1); throughout this section by anoperator on K[X] we shall mean a linear map U of the vector space K[X] (over K) into itself; since the monomials Xⁿ (n≥0) form a basis for this space, U is determined by the polynomials U(Xⁿ); specifically, if $f({\text{X}}) = \sum\limits_{k = 0}^\infty {{\lambda _k}} {{\text{X}}^k}$ with λ _k ∈K, then $U(f) = \sum\limits_{k = 0}^\infty {{\lambda _k}U} {\text{(}}{{\text{X}}^k})$