Some classes of permutation pentanomials

For each prime $p\ne 3$ and each power $q=p^k$, we present two large classes of permutation polynomials over $F_{q^2}$ of the form $X^{r}B\left(X^{q-1}\right)$ which have at most five terms, where $B\left(X\right)$ is a polynomial with coefficients in $\{1,-1\}$. The special case $p=2$ of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case $p>2$, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.