Exact divisibility by powers of the integers in the Lucas sequence of the first kind

Lucas sequence of the first kind is an integer sequence (U-n)(n >= 0) which depends on parameters a, b is an element of Z and is defined by the recurrence relation U-0 = 0, U-1 = 1, and U-n = aU(n-1) + bU(n-2) for n >= 2. In this article, we obtain exact divisibility results concerning U-n(k) for all positive integers n and k. This extends many results in the literature from 1970 to 2020 which dealt only with the classical Fibonacci and Lucas numbers (a = b = 1) and the balancing and Lucas-balancing numbers (a = 6, b = -1).