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

Abstract

Let (F_k,n)_n and (L_k,n)n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation a_n+1 = ka_n + a_n−1 with initial values F_k,0 = 0, F_k,1 = 1, L_k,0 = 2 and L_k,1 = k. In this paper, we characterize the p-adic orders ν_p(F_k,n) and ν_p(L_k,n) for all primes p and all positive integers k.