Free Martingale polynomials for stationary Jacobi processes

TL;DR

This paper extends the theory of free martingale polynomials to stationary free Jacobi processes with a broader parameter range, revealing new orthogonality properties and measures.

Contribution

It generalizes previous results on free martingale polynomials for stationary free Jacobi processes beyond the special case, introducing a multiplicative renormalization approach.

Findings

Polynomials are no longer orthogonal for most parameter values

Orthogonality measure is obtained via multiplicative renormalization

Results expand understanding of free Jacobi processes and their polynomial structures

Abstract

We generalize a previous result concerning free martingale polynomials for the stationary free Jacobi process of parameters $\lambda \in ]0.1], \theta = 1/2$. Hopelessly, apart from the case $\lambda = 1$, the polynomials we derive are no longer orthogonal with respect to the spectral measure. As a matter of fact, we use the multiplicative renormalization to write down the corresponding orthogonality measure.