On the sequences of $(q,k)$-generalized Fibonacci numbers

We consider a new family of recurrence sequences, the $(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of $k$-generalized Fibonacci numbers and generalized $k$-order Pell numbers. Further, we obtain the Binet formula and study the asymptotic behavior of the dominant root of the characteristic equation. The proof methods exploit pairs of characteristic polynomials which allow several auxiliary results.