Combinatorial proof of the gamma-deformed R-transform identities

Construct a combinatorial proof of the formal identities in Theorem \ref{thm:r_transform} relating the \gamma-quantized R-transform, the auxiliary coefficients c_n, and the moment sequence m_n, without relying on the Jack generating-function and Cherednik-operator method.

Background

Theorem \ref{thm:r_transform} establishes formal power-series relations between quantized \gamma-cumulants, auxiliary coefficients, and moments. Its proof uses the main law-of-large-numbers theorem, Jack generating functions, shifted Jack polynomials, gamma-function asymptotics, and Cherednik operators.

The authors explicitly identify the absence of an elementary or combinatorial proof as an unresolved problem. The requested proof would provide a more intrinsic explanation of the identities, analogous to combinatorial approaches used in related cumulant theories.

References

It should be noted that the main tool will be \cref{theo:main1}, which in turn was proved by employing Jack generating functions and Cherednik operators. It would be desirable to have an elementary proof along the same lines as Main Result~I. We leave finding a combinatorial proof as an open problem.

— Discrete $N$-particle systems at high temperature through Jack generating functions  (2502.13098 - Cuenca et al., 18 Feb 2025) in Paragraph immediately preceding the proof of Theorem \ref{thm:r_transform}, Section 6