The moments and cumulants of a real-valued random variable X is related via the cumulant-moment formula: $$m_n = \sum_{\pi\in P(n)}\prod_{B\in\pi}c_{|B|},\qquad\qquad(1)$$ where $m_n(X)=E(X^n)$ is the moment and $c_n(X)$ is the cumulant, and the sum runs over partitions of $[n]:=\{1,2\dots,n\}$, i.e., $\pi = \{B_1,B_2,\dots\}$, such that $[n]$ is the disjoint union of the $B_i$'s.
The moments and cumulants are also related via a simple formula: $$C(t) = \log M(t)=\log (E(\exp(tX))),\qquad\qquad(2)$$ where $$M(t)=\sum_{k=0}^\infty \frac{m_k}{k!}t^k\quad\text{and}\quad C(t)=\sum_{k=0}^\infty \frac{c_k}{k!}t^k$$ are the generating functions.
The moments $m$ and free cumulants $\kappa$ are related in a way similar to (1): $$m_n = \sum_{\pi\in NC(n)}\prod_{B\in\pi}\kappa_{|B|},\qquad\qquad(3)$$ where the sum runs over the non-crossing partitions.
So my question is, is there a nice formula for the "free cumulant generating function" $K(t)$ similar to (2)?$$K(t) = \sum_{k=0}^\infty \frac{\kappa_k}{k!}t^k.$$ Say, $$K(t)=\phi_2(E(\phi_1(tX)),$$ for some smooth functions $\phi_1$, $\phi_2$, or even for $\phi_2=\phi_1^{-1}$?
Thanks in advance!
Some refs:
J. Novak and P. ´Sniady, “What is...a free cumulant?"
J. A. Mingo and R. Speicher, Free Probability and Random Matrices, Springer, New York, 2017
The answer is yes. One can define the generating functions without the factorials as $$M(z)=\sum_{n=0}^\infty m_n z^n,\quad C(z)=\sum_{n=0}^\infty\kappa_n z^n.$$ Then your equation (3) is equivalent to (Speicher 1994) $$M(z)=C(zM(z)).$$