Solutions to the recurrence relation u n+1 = v n+1 + u n v n in terms of Bell polynomials

Motivated by time series analysis, we consider the problem of solving the recurrence relation u n+1 = v n+1 + u n v n for n ≠ 0 and u n, given the sequence v n. A solution is given as a Bell polynomial. When v n can be written as a weighted sum of nth powers, then the solution u n also takes this form. Mathematical subject classification: 33E99.