The p-Adic order of the k-Fibonacci and k-Lucas numbers

Abstract

Let (Fk,n)n and (Lk,n)n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation an+1 = kan + an−1 with initial values Fk,0 = 0, Fk,1 = 1, Lk,0 = 2 and Lk,1 = k. In this paper, we characterize the p-adic orders νp(Fk,n) and νp(Lk,n) for all primes p and all positive integers k.