Commutant of sum of two quasihomogeneous Toeplitz operators

A major open question in the theory of Toeplitz operator on the Bergman space of the unit disk of the complex plane is to fully characterize the set of all Toeplitz operators that commute with a given one. In [2], the second author described the sum $S=T_{e^{im\theta }f}+T_{e^{il\theta }g}$, where $f$ and $g$ are radial functions, that commutes with the sum $T=T_{e^{ip\theta }r(2M+1)p}+T_{e^{is\theta }r(2N+1)s}$ . It is proved that $S=cT$, where $c$ is a constant. In this article, we shall replace $r^{(2M+1)p}$ and $r^{(2N+1)s}$ by $r^n$ and $r^d$ respectively, with $n$ and $d$ in $\mathbb{N}$, and we shall show that the same result holds.