Let $(\mathcal A,\varphi)$ be a unital noncommutative probability space over $\mathbb C$: $\mathcal A$ is a unital complex algebra with multiplicative unit $1\in\mathcal A$, and $\varphi:\mathcal A\to\mathbb C$ is a linear functional satisfying $\varphi(1)=1$. Let $(\kappa_n)_{n\in\mathbb N}$ be the sequence of free cumulant functionals associated to $\varphi$, where for each $n\in\mathbb N$ the map $\kappa_n:\mathcal A^n\to\mathbb C$ is the $n$-linear functional defined by the moment-cumulant formula over noncrossing partitions. Then, for every $a\in \mathcal A$ and every $\lambda\in\mathbb C$,

\begin{align*} \kappa_1(a-\lambda 1)=\kappa_1(a)-\lambda. \end{align*}

Moreover, for every integer $n\ge 2$,

\begin{align*} \kappa_n(a-\lambda 1,\dots,a-\lambda 1)=\kappa_n(a,\dots,a). \end{align*}