Asymptotic behaviour of iterated modified 401-1401-1401-1401-2401-2401-2transforms on some slowly convergent sequences

LetL(α, r) denote the class of complex sequences (x _n) having an asymptotic expansion of type $$x_n \sim \sum\limits_{i \ge 0} {c_i n^{ - (\alpha + ri)} } , c_0 \ne 0,\alpha > 0,r > 0.$$ We describe the asymptotic behaviour of sequences obtained by applying to (x _n) some specific modified versions of the iterated Δ² transform, the θ₂ transform and some combinations of them. In this paper, we study the particular casesr=1 andr=1/2, which are the most useful in practice. The results are also valid for sequences (S _n) whose error sequence (x _n) defined byx _n=S _n?S, S=limS _n, belongs to someL(α, r).