Second order free cumulants: product, commutator, and anti-commutator

Given two second order free random variables $a$ and $b$, we study the second order free cumulants of their product $ab$, their commutator $ab-ba$, and their anti-commutator $ab+ba$. Let $(\kappa_n^a)_{n\geq 1}$ and $(\kappa_{n,m}^a)_{n,m\geq 1}$ denote the sequence of free cumulants of first and second order, respectively, of a random variable $a$ in a second order non-commutative probability space $(\mathcal{A},\varphi,\varphi^2)$. Given $a$ and $b$ two second order freely independent random variables, we provide formulas to compute each of the cumulants $(\kappa_{n,m}^{ab})_{n,m\geq 1}$, $(\kappa_{n,m}^{ab-ba})_{n,m\geq 1}$, and $(\kappa_{n,m}^{ab+ba})_{n,m\geq 1}$ in terms of the individual cumulants $(\kappa_{n}^{a})_{n\geq 1}$, $(\kappa_{n,m}^{a})_{n,m\geq 1}$, $(\kappa_{n}^{b})_{n\geq 1}$, and $(\kappa_{n,m}^{b})_{n,m\geq 1}$. For $n=m=1$ our formulas read: \begin{align*} \kappa_{1,1}^{ab} &= \kappa_{2}^{a}\kappa_{2}^{b} +\kappa_{1,1}^{a}(\kappa_{1}^{b})^2+\kappa_{1,1}^{b}(\kappa_{1}^{a})^2,\\ \kappa_{1,1}^{ab-ba} &= 2\kappa_{2}^{a}\kappa_{2}^{b},\\ \kappa_{1,1}^{ab+ba} &= 2\kappa_{2}^{a}\kappa_{2}^{b} +4\kappa_{1,1}^{a}(\kappa_{1}^{b})^2+4\kappa_{1,1}^{b}(\kappa_{1}^{a})^2. \end{align*} In general, our formulas express the cumulants $\kappa_{n,m}^{ab}$, $\kappa_{n,m}^{ab-ba}$, and $\kappa_{n,m}^{ab+ba}$ as sums indexed by special subsets of non-crossing partitioned permutations. The formulas for the commutator and anti-commutator where not studied before, while the formula for the product was only known in the case the where the individual second order free cumulants vanish. As an application, we compute explicitly the cumulants of the anti-commutator and product of two second order free semicircular variables.