Permutations of finite fields for check digit systems (Q1960224)

From the authors' abstract: Let \(q\) be a prime power and \(n\) a positive divisor of \(q-1\). We prove an asymptotic formula for the number of polynomials \[ f(X)=\frac{a-b}{n}\bigg(\sum_{j=1}^{n-1} X^{j(q-1)/n}\bigg)X+\frac{a+b(n-1)}{n}\,X\in\mathbb{F}_q[X] \] such that the polynomials \(f(X)\), \(f(X)\pm X\), and \(f(f(X))\pm X\) are all permutation polynomials over \(\mathbb{F}_q\).