  Permutation polynomials over finite fields are an interesting subject due to
their important applications in the areas of mathematics and engineering. In
this paper we investigate the trinomial $f(x)=x^{(p-1)q+1}+x^{pq}-x^{q+(p-1)}$
over the finite field $\mathbb{F}_{q^2}$, where $p$ is an odd prime and $q=p^k$
with $k$ being a positive integer. It is shown that when $p=3$ or $5$, $f(x)$
is a permutation trinomial of $\mathbb{F}_{q^2}$ if and only if $k$ is even.
This property is also true for more general class of polynomials
$g(x)=x^{(q+1)l+(p-1)q+1}+x^{(q+1)l+pq}-x^{(q+1)l+q+(p-1)}$, where $l$ is a
nonnegative integer and $\gcd(2l+p,q-1)=1$. Moreover, we also show that for
$p=5$ the permutation trinomials $f(x)$ proposed here are new in the sense that
they are not multiplicative equivalent to previously known ones of similar
form.
