Deformations of the Taylor Formula

By a deformation of the integers we mean a sequence x = {xn, n ∈ N} of polynomials in one or more variables and with integral coefficients, having the property that there exists some value q0 of the variables such that ∀n ∈ N, xn(q0) = n. The quantum integers xn = ∑n−1 l=0 q l are a typical example of a deformation of the integers. Another example is given by the version of the Chebyshev polynomials defined by xn(cos(θ)) = sin(nθ) sin(θ) . In this note we consider some deformations of the factorial function and of the binomial coefficients that are induced by such deformations of the integers. This situation can be interpreted as a deformation of the Taylor formula, as explained below. Given a polynomial P of degree n with complex coefficients, the Taylor expansion at some point X gives