Subscribe to Events

Many important classical families of orthogonal polynomials, such as Jacobi, Laguerre, Legendre, etc., arise via specialization from some hypergeometric series or from a q-analog (the so-called basic series). The Askey scheme is a hierarchical list of some 48 such families, and it is natural to ask if this list is complete.

In joint work with Joseph Bernstein and Dmitry Gourevitch (https://arxiv.org/abs/2401.14715), we have axiomatized some part of the Askey scheme, which we refer to as families of Jacobi type, and discovered 2 new families.

In this lecture I will describe the q-analog of the above result, namely the definition and classification of polynomials of basic Jacobi type. It turns out that in this setting there are 4 new families beyond the Askey scheme.

We are in the process of extending our results to obtain a complete classification of hypergeometric and basic orthogonal polynomials. Our expectation is that we will recover the full Askey scheme along with a small number of new families.