Hermite Orthogonality from Bispectral Wave Functions of the KP Hierarchy, Alex Kasman

[12/12/2025] Hermite Orthogonality from Bispectral Wave Functions of the KP Hierarchy
Alex Kasman, College of Charleston, USA

Abstract:
The Classical Hermite Polynomials (CHPs) and the Exceptional Hermite Polynomials (EHPs) provide many examples of functions that are orthogonal with respect to the Hermite inner product. The KP Hierarchy is a collection of infinitely-many compatible non-linear PDEs that can be solved exactly and have "particle-like"soliton solutions. At first, it might seem that these two topics are unrelated. However, in 2020 we showed that the generating functions of each family of EHPs happen to be KP wave functions. In this talk, I will report on new results that both generalize and explain this connection between KP and EHPs. In particular, using the dynamical equations of the KP Hierarchy, we will show that any coefficient from the power series expansion of a bispectral KP Wave Function is Hermite orthogonal to all but a finite number of coefficients from its bispectral dual. The wave function itself (evaluated at a different time) contains the information to determine precisely when they will be orthogonal. As a special case, the orthogonality of the CHPs and EHPs can be independently rederived from this soliton theoretic construction. This procedure has be generalized to produce matrix functions that are Hermite orthogonal and we are working on extending it to work with other classical inner products.