Irreducible factorization of translates of reversed Dickson polynomials over finite fields

Let Fq be a field of q elements, where q is a power of an odd prime. Fix n=(q+1)/2. For each s∈Fq, we describe all the irreducible factors over Fq of the polynomial gs(y):=yn+(1−y)n−s, and we give a necessary and sufficient condition on s for gs(y) to be irreducible.