Hello everyone, I have the following problem.

Let (u_n)_{n>=0} be a bounded sequence of non-negative integers.

Set (v_n)_{n>=0} as the sequence such that v_0=1 and v_n is the sum of terms u_k v_{n-k} for k between 1 and n

(v_n = u_0 v_{n-1} + u_1 v_{n-2} + ... + u_{n-1} v_0).

It is well known that if (u_n) has only finitely many non zero terms, then (v_n) is given by a linear recurrence, and using a trick this is still true if (u_n) is periodic. But what happens if (u_n) is not periodic ?

At the moment I have already proven that there exists a real number lambda such that v_n /lambda^n lies in the interval [0;1], that there is a lim inf >0 and a lim sup <1. But is it possible to say more? (somebody told me to use power series but I don't know exactly what do to...)

Thanks by advance for your comments, and sorry for my poor English :/