The asymptotic behavior of a family of sequences via Tauberian theorems

We study the asymptotic behavior of a family of sequences defined by the fo llowing nonlinear induction relation c(sigma) = 1 and c(n) := Sigma (k)(j=1 ) r(j)c([n/mj]) + Sigma (k)(j=k+1) r(j)c([(n+1)1/mj]-1) for n greater than or equal to 1, where the r(j) are real positive numbers and m(j) are intege rs greater than or equal to 2. Depending on the fact that Sigma (k)(j=1) r( j) is greater or lower than 1, we prove that c(n)/n(alpha) or c(n)/(ln n)(a lpha) goes to some finite limit for some explicit a. Our study is based on Tauberian theorems and extends a result of Erdos et al. (C) 2001 Academic P ress.