Physical Math Seminar: From Krawtchouk to Hahn via Jacobi and some other applications of differential operators with polynomial coefficients

报告人： Mikhail Tyaglov (Saint Petersburg State University)

Time: 2025-12-11 14:00-16:00
Location: 1610, SIMIS
Zoom Meeting ID: 895 1638 2738 Passcode: SIMIS

摘要：

Consider a linear differential operator of the form
(1) L_r u := \sum_{j=0}^{r} Q_j(z) \frac{d^j u(z)}{dz^j},
where the coefficient Q_j is a polynomial of degree n_j, j = 0,1,…,r, and Q_0(0)=0. For certain choices of coefficients, the operator L_r has polynomial eigenfunctions. In particular, the infinite systems of classical orthogonal polynomials are eigenfunctions of certain specializations of L_2.

We are interested in describing operators of the form (1) that have at least one polynomial eigenfunction, as well as in studying their properties. The first goal of the talk is to present our approach to studying and recognizing such operators. The second goal is to survey various known (finite and infinite) systems of polynomial eigenfunctions of operators of type (1) of order 1 and 2.

Additionally, we will present a variation of our method which turns out to be connected with the so-called tridiagonalisation of differential operators introduced by E. Koelink and M. Ismai [2]. In particular, we reveal an interesting relation between the Krawtchouk and Hahn polynomials relying on certain properties of the Jacobi polynomials.

Moreover, we survey some recent results on the generalisation of the Bochner theorem [1] and give an interpretation of those results from our approach.

The talk is based on joint work with Alexander Dyachenko.

References:
[1] L. M. Anguas, D. Barrios Rolanía, On polynomial solutions of certain finite order ordinary differential equations, Linear Algebra Appl. 721 (2022), 122–148.
[2] M. Ismail, E. Koelink, Spectral properties of operators using tridiagonalization, Analysis and Applications 10(3), 2012, 327–343.