Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials

What the math gives to ML

The paper provides a constructive family of sparse delayed recurrences whose characteristic polynomials have exact (r+1)-fold rotational symmetry and whose eigenvalues lie on an (r+1)-ray star. This structure can be transferred into neural state-space or polynomial-filter layers as a parameter-efficient alternative to unconstrained deep recurrence: periodic positive coefficients impose predictable spectral geometry while reducing the number of learned parameters. A second transferable tool is diagonal similarity scaling of the associated Hessenberg operator, which gives a cheap way to control non-normal amplification through an induced infinity-norm bound. The strongest first experiments are small recurrent layers and graph polynomial filters, comparing stability and accuracy against untied delayed residual blocks at equal parameter count.