Hodge filtration and Hodge ideals for $\mathbb{Q}$-divisors with weighted homogeneous isolated singularities or convenient non-degenerate singularities

We give an explicit formula for the Hodge filtration on the $\mathscr{D}_X$-module $\mathcal{O}_X (*Z) f^{1-\alpha}$ associated to the effective $\mathbb{Q}$-divisor $D = \alpha \cdot Z$, where $0 \lt \alpha \leq 1$ and $Z = (f = 0)$ is an irreducible hypersurface defined by $f$, a weighted homogeneous polynomial with an isolated singularity at the origin. In particular this gives a formula for the Hodge ideals of $D$. We deduce a formula for the generating level of the Hodge filtration, as well as further properties of Hodge ideals in this setting. We also extend the main theorem to the case when $f$ is a germ of holomorphic function that is convenient and has non-degenerate Newton boundary.