Extended procedures for extrapolation to the limit

In this paper, new procedures for the extrapolation to the limit of slowly convergent sequences and functions are proposed. They are based on the notions of error estimates and annihilation operators. We obtain generalizations of the discrete and confluent E-transformation, which are the most general sequence and function transformations known so far. It is shown that many transformations studied in the literature are included in our formalism. Particular cases of these procedures are discussed. Then, several extensions of the E-algorithm are given. Finally, the procedure $\theta$ is applied to our procedures to produce new procedures for extrapolation to the limit.