Exceptional orthogonal polynomials and Wronskians

My research explores Wronskian polynomials which appear in the field of exceptional orthogonal polynomials and also play a key role in the rational solutions of Painlevé equations. Both research fields are rapidly gaining interest and their connections to Wronskian polynomials are increasingly studied. The goal of my research is to explain these connections and describe their valuable features.

Exceptional orthogonal polynomials were introduced in 2009 as an extension of the classic orthogonal polynomials. Their characteristic feature is that the sequence of polynomials has finitely many gaps in their degree sequence, meaning that there is no polynomial for a finite number of degrees. The first part of my thesis covers a systematic construction of exceptional orthogonal polynomials labeled by partitions, as well as numerous results that are expressed via these partitions. The exceptional Laguerre polynomials and the exceptional Jacobi polynomials in particular are analyzed in detail, and the asymptotic properties of their zeros are derived. Exceptional Hermite polynomials are covered as well, these polynomials are described within the class of Wronskian Hermite polynomials for which different recurrence relations are determined. These results are then translated into recurrence relations for Wronskian Laguerre polynomials, and generalized to results for Wronskian Appell polynomials. This generalization is established via a connection with symmetric function theory, which emphasizes that the use of partitions not only makes the labeling elegant, but also that combinatorial results translate into remarkable properties for these polynomials. For example, it is shown that Wronskian Hermite polynomials have integer coefficients via this connection. Next, these coefficients are examined in more detail where the combinatorial concepts of cores and quotients are crucial. The fact that these coefficients can be traced by means of the characters of the irreducible representations of the symmetric group shows once again that the combinatorial framework gives many intriguing results.