  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.
