Let $(\mathcal A,\varphi)$ be a noncommutative probability space, and let $\kappa_n:\mathcal A^n\to\mathbb C$, for $n\geq 1$, be the free cumulants associated to $\varphi$ by the Speicher moment-cumulant formula over noncrossing partitions. For a block $V=\{i_1<\cdots<i_r\}\subseteq\{1,\ldots,n\}$, the block cumulant is evaluated in the natural increasing order as $\kappa_r(a_{i_1},\ldots,a_{i_r})$. Then, for every $a_1,a_2,a_3,a_4\in\mathcal A$, the first four identities are

\begin{align*} \varphi(a_1)=\kappa_1(a_1). \end{align*}

\begin{align*} \varphi(a_1a_2)=\kappa_2(a_1,a_2)+\kappa_1(a_1)\kappa_1(a_2). \end{align*}

\begin{align*} \varphi(a_1a_2a_3)=\kappa_3(a_1,a_2,a_3)+\kappa_2(a_1,a_2)\kappa_1(a_3)+\kappa_2(a_1,a_3)\kappa_1(a_2)+\kappa_2(a_2,a_3)\kappa_1(a_1)+\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3). \end{align*}

\begin{align*} \varphi(a_1a_2a_3a_4)=\kappa_4(a_1,a_2,a_3,a_4)+\kappa_3(a_1,a_2,a_3)\kappa_1(a_4)+\kappa_3(a_1,a_2,a_4)\kappa_1(a_3)+\kappa_3(a_1,a_3,a_4)\kappa_1(a_2)+\kappa_3(a_2,a_3,a_4)\kappa_1(a_1)+\kappa_2(a_1,a_2)\kappa_2(a_3,a_4)+\kappa_2(a_1,a_4)\kappa_2(a_2,a_3)+\kappa_2(a_1,a_2)\kappa_1(a_3)\kappa_1(a_4)+\kappa_2(a_1,a_3)\kappa_1(a_2)\kappa_1(a_4)+\kappa_2(a_1,a_4)\kappa_1(a_2)\kappa_1(a_3)+\kappa_2(a_2,a_3)\kappa_1(a_1)\kappa_1(a_4)+\kappa_2(a_2,a_4)\kappa_1(a_1)\kappa_1(a_3)+\kappa_2(a_3,a_4)\kappa_1(a_1)\kappa_1(a_2)+\kappa_1(a_1)\kappa_1(a_2)\kappa_1(a_3)\kappa_1(a_4). \end{align*}