Relation to finite free probability

Determine whether the formal power-series equations relating the \gamma-quantized R-transform, the coefficients c_n, and the moment sequence m_n have a relation to Finite Free Probability.

Background

The paper derives two formal generating-function identities connecting quantized \gamma-cumulants, an auxiliary sequence of coefficients, and moments. It compares these identities with analogous formulas in continuous beta-ensemble theory and with cumulant-based approaches to Finite Free Probability.

Although related identities are known in continuous settings and the paper observes structural similarities, it does not establish whether the discrete \gamma-deformed equations belong to or interact with Finite Free Probability. The question is therefore explicitly left unresolved.

References

Whether our equations~eq:1--eq:2 have any relation to Finite Free Probability is an open problem.

— Discrete $N$-particle systems at high temperature through Jack generating functions  (2502.13098 - Cuenca et al., 18 Feb 2025) in Remark following Theorem \ref{thm:r_transform}, Section 6 (The gamma-deformed R-transform and moment generating function)