The asymptotic behavior of a family of sequences via Tauberian theorems (Q5945102)

\textit{P. Erdős}, \textit{A. Hildebrand}, \textit{A. Odlyzko}, \textit{P. Pudaite} and \textit{B. Reznick} [The asymptotic behavior of a family of sequences, Pac. J. Math. 126, No. 2, 227-241 (1987; Zbl 0558.10010)] have investigated the asymptotic behaviour of certain recursively defined sequences. Let be given \(k\in \mathbb{N}\), \(r_j>0\), \(2\leq m_j\in \mathbb{N}\) \((1\leq j\leq k)\) such that \(\log m_i/ \log m_j\notin \mathbb{Q}\) for some pair \((i,j)\) to define the recursion \(a_0=1\), \(a_n=\sum^k_{j=1} r_ja_{[n/m_j]}\), \(n\geq 1\). It was shown under additional conditions that \(a_n\sim cn^\alpha\) with an explicit constant \(c\) and some constant \(\alpha\) solving \(\sum^k_1 r_jm_j^{-\alpha} =1\).NEWLINENEWLINENEWLINEIt is the aim of the present paper to give a similar result for a more general recursion of type: NEWLINE\[NEWLINEc_0=1,\quad c_n= \sum^k_{j=1} r_jc_{[n/m_j]} +\sum^K_{j=k +1} r_jc_{[(n+1)^{1/m_j}]-1}, \quad n\geq 1.NEWLINE\]NEWLINE The main tools for the proof are versions of the Ikehara Tauberian theorem for Laplace transforms, which are discussed in the paper as well, in order to get back the asymptotic behaviour of the sequence \((c_n)\) from that of the Mellin transform of \(c_{[x]}\).