The distribution of polynomials in monotone-independent elements

Building on the work of Arizmendi and Celestino (2021), we derive the *-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers α and β, we derive explicitly the *-distribution of pα, β = αab + βba whenever a and b are monotone or infinitesimally monotone independent elements. This encompasses both cases of the commutator and anti-commutator. This approach can be pushed to study more general polynomials. As applications, we derive the limiting distribution with respect to the partial trace of polynomials in a certain class of random matrices.