  We show the existence of many infinite classes of permutations over finite
fields and bent functions by extending the notion of linear translators,
introduced by Kyureghyan [12]. We call these translators Frobenius translators
since the derivatives of $f : F_{p^n} \rightarrow F_{p^k}$, where $n = rk$, are
of the form $f(x + u\phi) - f(x) = u^{p^i}b$, for a fixed $b \in F_{p^k}$ and
all $u \in F_{p^k}$, rather than considering the standard case corresponding to
$i = 0$. This considerably extends a rather rare family {f} admitting linear
translators of the above form. Furthermore, we solve a few open problems in the
recent article [4] concerning the existence and an exact specification of $f$
admitting classical linear translators, and an open problem introduced in [9]
of finding a triple of bent functions $f_1, f_2, f_3$ such that their sum $f_4$
is bent and that the sum of their duals $f_1* +f_2* +f_3* +f_4* = 1$. Finally,
we also specify two huge families of permutations over $F_{p^n}$ related to the
condition that $G(y) = -L(y)+(y+\delta)^s -(y+\delta)^{p^ks}$ permutes the set
$S =\{\beta \in F_{p^n} : Tr^n_k(\beta) = 0\}$, where $n = 2k$ and $p > 2$.
Finally, we offer generalizations of constructions of bent functions from [16]
and described some new bent families using the permutations found in [4].
