  In this paper, we consider the new family of recurrence sequences of
$(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the
well-known sequences of $k$-generalized Fibonacci numbers and generalized
$k$-order Pell numbers. We shall obtain a Binet-style formula and study the
asymptotic behavior of dominant root of characteristic equation. Moreover, we
shall prove some auxiliary results about these sequences. In particular, we
characterize the first $(q,k)$-generalized Fibonacci numbers in terms of binary
sequences.
