The Asymptotic Behavior of a Family of Sequences via Tauberian Theorems

We study the asymptotic behavior of a family of sequences defined by the following nonlinear induction relation c₀ = 1 and c_n ∑^k_{j = 1} r_jc_[n/mj] + ∑^k_{j = k + 1} r_jc_{[(n + 1)^1/mj] − 1} for n ≥ 1, where the r_j are real positive numbers and m_j are integers greater than or equal to 2. Depending on the fact that ∑^k_{j = 1} r_j is greater or lower than 1, we prove that c_n/n^α or c_n/(ln n)^α goes to some finite limit for some explicit α. Our study is based on Tauberian theorems and extends a result of Erdös et al.