New permutation polynomials over ${\mathbb{F}}_{q^2}$

In this paper, we propose a new method to obtain new permutation polynomials over ${\mathbb{F}}_{q^2}$. Using this method, we extend many known permutation polynomials, which take the form ${\sum }_i(x^q-x+\delta {)}^{s_i}+L(x)$, where $L(x)$ is a $q$-polynomial over ${\mathbb{F}}_q$ and $\delta \in {\mathbb{F}}_{q^2}$. We also present an alternative approach for constructing permutation polynomials of the form $x+\gamma T{r}_q^{q^d}(x^{q+1}+x^{2q+2})$ for the cases where $q=2^m$, $2\nmid d$ and $T{r}_q^{q^d}(x)=x+x^q+\cdots +x^{q^{d-1}}$.