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In this paper, we construct two classes of permutation polynomials over$\mathbb{F}_{q^2}$ with odd characteristic from rational R\'{e}dei functions. Acomplete characterization of their compositional inverses is also given. Thesepermutation polynomials can be generated recursively. As a consequence, we cangenerate recursively permutation polynomials with arbitrary number of terms.More importantly, the conditions of these polynomials being permutations arevery easy to characterize. For wide applications in practice, several classesof permutation binomials and trinomials are given. With the help of a computer,we find that the number of permutation polynomials of these types is verylarge.

A Recursive Construction of Permutation Polynomials over $\mathbb{F}_{q^2}$ with Odd Characteristic from R\'{e}dei Functions