Post-Hopf algebra in non-commutative probability theory

We study $\mathcal{O}$-operators and post-Lie products over the same Lie algebra compatible in a certain sense. We prove that the group product corresponding to the formal integration of the Lie algebra, which is adjacent to the sum of two compatible post-Lie products, can be factorized in a way reminiscent of the classical Semenov-Tian-Shanskii factorization. In the second part, we explore applications in non-commutative probability. We introduce new transforms that facilitate the computation of conditionally free and conditionally monotone multiplicative convolutions involving operator-valued non-commutative distributions.