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

Abstract

For d ≥ 2, denote by Pd(x) the polynomial over 𝔽q given by [Formula: see text]. We explicitly find the number of solutions in 𝔽q of the polynomial equation Pd(x) = 0 in terms of special values of dFd-1 and d-1Fd-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 n+1Fn Gaussian hypergeometric series for n > 2.