Congruences Involving Sums of Ratios of Lucas Sequences

Given a pair (U-t) and (V-t) of Lucas sequences, we establish various congruences involving sums of ratios V-t/U-t. More precisely, let p be a prime divisor of the positive integer m. We establish congruences, modulo powers of p, for the sum Sigma V-t/U-t, where t runs from 1 to r(m), the rank of m, and r(q) inverted iota t for all prime factors q of m.