The concepts of free cumulants and free moments are indispensably related to the idea of freeness introduced by Voiculescu [Voiculescu, D., Proc. Conf., Buşteni/Rom., Lect. Notes Math. 1132(1985), pp. 556-588] and studied further within Free probability theory. Free probability theory is of great importance for both the developing mathematical theories as well as for problem solving methods in engineering.
The goal of this paper is to present theoretical framework for free cumulants and moments, and then prove a new free cumulant-moment relation formula. The existing relations between these objects will be given. We consider as drawback that they require the combinatorial understanding of the idea of non--crossing partitions, which has been considered by Speicher [Speicher, R., Math. Ann., 298(1994), pp. 611-628] and then widely studied and developed by Speicher and Nica [Nica, A. and Speicher, R.: Lectures on the Combinatorics of Free Probability, Cambridge University Press, Cambridge, United Kingdom, 2006]. Furthermore, some formulations are given with additional use of the Möbius function. The recursive result derived in this paper does not require introducing any of those concepts, instead the calculations of the Stieltjes transform of the underlying measure are essential.
The presented free cumulant--moment relation formula is used to calculate cumulants of degree 1 to 5 as a function of the moments of lower degrees. The simplicity of the calculations can be observed by a comparison with the calculations performed in the classical way using non-crossing partitions. Then, the particular example of non-commutative space i.e., space of p×p matrices X=(X_{ij})_{ij}, where X_{ij} has finite moments, equipped with functional E(TrX)∕p is investigated.