Acceleration property for the E-algorithm and an application to the summation of series

The E-algorithm is the most general sequence transformation actually known, since it contains as particular cases almost all the sequence transformations discovered so far : Richardson polynomial extrapolation , Shanks' transformation , summation processes , Germain-Bonne transformation , Levin's generalized transformations , the process p and rational extrapolation. In 10] some results concerning the columns of the E-algorithm were proved. In this paper , by adding conditions about determinants , we prove that the diagonal of this algorithm also accelerates the convergence of the initial sequence .