ON THE POLYNOMIAL xd + ax plus b OVER Fq AND GAUSSIAN HYPERGEOMETRIC SERIES

For d >= 2, denote by P-d(x) the polynomial over F-q given by P-d(x) = x(d) + ax + b. We explicitly find the number of solutions in F-q of the polynomial equation P-d(x) = 0 in terms of special values of F-d(d-1) and F-d-1(d-2) Gaussian hypergeometric series with characters of orders d and d - 1 as parameters. This solves a problem posed by K. Ono (see p. 204 in [Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, No. 102 (American Mathematical Society, Providence, RI, 2004)]) on special values of F-n+1(n) Gaussian hypergeometric series for n > 2.