An extension of Lucas' theorem

Let $p$ be a prime. A famous theorem of Lucas states that $\binom {mp+s}{np+t}\equiv \binom mn\binom st (\operatorname{mod} p)$ if $m,n,s,t$ are nonnegative integers with $s,t<p$. In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values $0$ and $1$.