Mixed Bruce-Roberts numbers

We extend the notion of $\mu^*$-sequence and Tjurina number of functions to the framework of Bruce-Roberts numbers, that is, to pairs formed by the germ at $0$ of a complex analytic variety $X\subseteq \mathbb C^n$ and a finitely $\mathcal R(X)$-determined analytic function germ $f:(\mathbb C^n,0)\to (\mathbb C,0)$. We analyze some fundamental properties of these numbers.