On the uniqueness of the multirational blossom

The multirational blossom of order k and degree −n of a k-times differentiable function $f\left(t\right)$ is defined as a multivariate function $f(u_1,\dots ,u_k/v_1,\dots ,v_{k+n})$ characterized by four axioms: bisymmetry in the u and v parameters, multiaffine in the u parameters, satisfies a cancellation property and reduces to $f\left(t\right)$ along the diagonal. The existence of a multirational blossom was established in Goldman (1999a) by providing an explicit formula for this blossom in terms of divided differences. Here we show that these four axioms uniquely characterize the multirational blossom. We go on to introduce a homogeneous version of the multirational blossom. We then show that for differentiable functions derivatives can be computed in terms of this homogeneous multirational blossom. We also use the homogeneous multirational blossom to convert between the Taylor bases and the negative degree Bernstein bases.