Let p be a prime number, Zp the ring of the p-adic integers, Qp the field of the p-adic numbers and K a complete valued field, extension of Qp. Let C(Zp,K) be the Banach algebra of continuous functions from Zp to K equipped with the supremum norm. K. Conrad has given a q-analogue of Mahler's expansion for q 2 K, |q − 1| < 1. We use the techniques of umbral calculus to establish a bijective correspondence, on one hand: between a class of continuous functions which are orthonormal q-bases and a class of linear continuous operators which commute with 1 such that 1(f)(x) = f(x + 1); on the other hand between a class of orthogonal q-bases of C(Zp,K) and a class of linear continuous operators which commute with the Jackson q-derivation. We give a realization of the quantum plane and Weyl quantum algebra of two generators, in the form of concrete operators algebras. We do some calculus of norms of these operators and exhibit an interesting orthogonal family for the quantum Weyl algebra. We provide a necessary and sufficient conditions on the coefficients of the q-expansion for a continuous function to be strictly differentiable, first when q is not a root of unity and after when q is a primitive pN-th root of unity. As an application we give a q-expansion of the Volkenborn integral.
