  Permutation rational functions over finite fields have attracted much
attention in recent years. In this paper, we introduce a class of permutation
rational functions over $\mathbb F_{q^2}$, whose numerators are so-called
$q$-quadratic polynomials. To this end, we will first determine the exact
number of zeros of a special $q$-quadratic polynomial in $\mathbb F_{q^2}$, by
calculating character sums related to quadratic forms of $\mathbb
F_{q^2}/\mathbb F_q$. Then given some rational function, we can demonstrate
whether it induces a permutation of $\mathbb F_{q^2}$.
