Ridge polynomials (on the disk)?

[MikaelSlevinsky]

These look like P_{n,k}(x,y) = U_n(x * cos(k*pi/(n+1)) + y * sin(k*pi/(n+1))), orthogonal on the unweighted unit disk.

Sparse differentiation is trivial (using ultraspherical generalizations), but the Jacobi operators are not obvious.

[MikaelSlevinsky]

https://www.math.auckland.ac.nz/~waldron/Preprints/Disc-Polys/discpolys.pdf

[dlfivefifty]

Yes I know this family from Dunkl and Xu, it’s cool and doesn’t require any high order OPs