Given an integer base b ≥ 2, a hyper b-ary representation of a positive integer n is a representation of n as a linear combination of nonnegative powers of b, with integer coefficients between 0 and b. We use a system of recurrence relations to define a sequence of polynomials in b variables and with b parameters, and we show that all hyper b-ary representations of n are characterized by the polynomial with index n+1. This extends a recent result of Defant on the number of hyper b-ary representations based on a b-ary analogue of Stern's diatomic sequence. The polynomials defined here extend this numerical sequence, and they can be seen as generalized b-ary Stern polynomials.