p-adic Gamma function and the polynomials xd+ax+b and xd+axd−1+b over Fq

In [14], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function. He expressed the trace of Frobenius of elliptic curves in terms of special values of G22[⋯]. For d⩾2, we establish four different expressions for the number of distinct zeros of the polynomials xd+ax+b and xd+axd−1+b over Fq in terms of special values of the function Gd−1d−1[⋯].