A combinatorial identity for the Jacobian of $t$-shifted invariants

Let $\mathfrak{g}$ be a simple Lie algebra. There are classical formulas for the Jacobians of the generating invariants of the Weyl group of $\mathfrak{g}$ and of the images under the Harich-Chandra projection of the generators of $ZU(\mathfrak{g})$. We present a modification of these formulas related to Takiff Lie algebras.