Two types of permutation polynomials with special forms

Let $q$ be a power of a prime and $\mathbb{F}q$ be a finite field with $q$ elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form $cx-x^s + x^{qs}$ over $\mathbb{F}{q^2}$, and investigate the relationship between this type of permutation polynomials with that of the form $(x^{q-x+\delta)s+cx$.} Based on this relation, many classes of permutation trinomials having the form $(x^{q-x+\delta)s+cx$} without restriction on $\delta$ over $\mathbb{F}_{q^2}$ are derived from known permutation trinomials having the form $cx-x^s + x^{qs}$.