Non-polynomial divided differences and B-spline functions

Starting with the interpolation problem, we define non-polynomial divided differences recursively as a generalization of classical divided differences. We also derive the identities and the properties of these non-polynomial divided differences such as symmetry and Leibniz formula which is a main tool in the derivation of B-spline recurrence relations. Defining a novel variant of the truncated power function, we express non-polynomial B-splines explicitly in terms of non-polynomial divided differences of this truncated power function.