Given an integer b ≥ 2, a well studied concept of a b-ary partition function of a positive integer n counts the number of representations of n as sums of powers of b, with each power occurring up to λ times, for a fixed λ ≥ 1. In this paper we introduce and study a multivariable polynomial sequence that reduces to the restricted b-ary partition function when all variables are taken to be 1. In particular, we show that this polynomial sequence characterizes all restricted b-ary partitions for each n, generalizing previous results on hyperbinary and hyper b-ary representations. All this will follow from considering more general concepts of restricted b-ary partition functions.