Add parameters while preserving q-Gaussian solvability

Determine how to introduce additional independent parameters into the q-Gaussian matrix model while retaining an explicitly solvable model and obtaining a quantum curve with generic coefficients.

Background

The generic two-parameter Al-Salam–Carlitz deformation of the q-Gaussian weight produces a quantum curve whose coefficients remain constrained: only two combinations of coefficients are arbitrary. The authors explicitly state that they do not know how to enlarge the parameter space without losing explicit solvability.