Multiplicative dependence of k-Fibonacci numbers with the Fibonacci, Lucas, and Pell sequences

The k –generalized Fibonacci sequence (Fₘ⁽ᵏ⁾)m≥ 2-k ( F m ( k ) ) m ≥ 2 - k is the linear recurrent sequence of order k whose first k terms are 0, … , 0, 1 0 , … , 0 , 1 and each term afterwards is the sum of the preceding k terms. The case $$k=2$$ k = 2 corresponds to the well known Fibonacci sequence. In Gómez and Luca (Lith. Math. J. 56(4):503–517, 2016), the multiplicative independence between terms of the same k -generalized Fibonacci sequence was studied. In this paper, we find all the multiplicative dependent pairs (Fₘ⁽ᵏ⁾,uₙ) ( F m ( k ) , u n ) where uₙ u n is a Fibonacci, a Lucas or a Pell number.