A q-analogue of Mahler Expansions

Since the work of K. Mahler about forty years ago, the binomial coefficient polynomials have played an important role in working with continuous functions on the p-adic integers which take p-adic values. The q-analogue of binomial coefficients will be shown to admit a similar role when q is a p-adic variable close to 1. Mahler's construction is recovered at q = 1, and several properties of his construction will be extended to the q-analogue.