Analysis Seminar

Spring 2026

For the Spring 2026 semester we will meet virtually on zoom at 11:00 am on Fridays. For more information contact Mourad Ismail or Xiang-Sheng Wang.

1. 16 January 2026 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
   Two classes of Chebyshev-like polynomials
   Karl Dilcher
   Dalhousie University
   Abstract: In this talk, I present two different variants of the classical Chebyshev polynomials. In the first part, a new polynomial sequence is obtained by altering the recurrence relation. Among other properties, we obtain results on their irreducibility and their zeros. We then study the $2\times 2$ Hankel determinants of these polynomials, which also have interesting zero distributions. Furthermore, if these polynomials are split into two halves, we still get meaningful properties. In the second part, a pair of polynomial sequences is obtained by altering the well-known functional equation of the Chebyshev polynomials of both kinds. We derive numerous properties of these new polynomials, including explicit expansions, differential equations, recurrence relations, generating functions, discriminants, irreducibility results, and their zeros. We also consider some related polynomial sequences and their properties. (Joint work with Ken Stolarsky, Univeristy of Illinois, and Maciej Ulas, Jagiellonian University in Krakow, Poland).
2. 23 January 2026 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
   Matrix and scalar Gegenbauer polynomials
   Erik Koelink
   Radboud Universiteit
   Abstract: After a brief recap of matrix orthogonal polynomials, we recall the matrix Gegenbauer polynomials for arbitrary size. We show that there exists a symmetric version of these polynomials, giving rise to various properties for these polynomials. Moreover, we present an explicit expression of the matrix Gegenbauer polynomials in terms of scalar Gegenbauer polynomials and vice versa. We end with a discussion of some of the entries of the symmetric matrix Gegenbauer polynomials and some open questions. The talk is based on a joint paper with Wadim Zudilin and Pablo Rom\'an in Pacific J Math 2025.
3. 30 January 2026 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
   Bailey pairs and quantum q-series identities
   Jeremy Lovejoy
   Université Paris Cité
   Abstract: In the first part of this talk I will survey classical q-hypergeometric series as functions inside the unit disk along with the classical question: What kinds of identities do these series satisfy and what are the applications of these identities in number theory, combinatorics, and beyond? In the second part of the talk I will discuss recent work on the same question but for q-hypergeometric series at roots of unity, with a focus on finding a path to establishing quantum modularity. A key role will be played throughout by Bailey pairs. Some of this work is joint with Jehanne Dousse and Amanda Folsom.
4. 6 February 2026 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
   Orthogonal Polynomials and the Associated Jacobi Operator
   Christian Berg
   University of Copenhagen, Denmark
   We consider the Jacobi operator $(T,D(T))$ associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many non-zero terms. It is well-known that it is symmetric with deficiency indices $(1,1)$. For a complex number $z$ let $\mathfrak{p}_z, \mathfrak{q}_z$ denote the square summable sequences $(p_n(z))$ and $(q_n(z))$ corresponding to the orthonormal polynomials $p_n$ and polynomials $q_n$ of the second kind. We determine whether linear combinations of $\mathfrak{p}_u, \mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ for $u,v\in\C$ belong to $D(T)$ or to the domain of the self-adjoint extensions of $T$ in $\ell^2$. The results depend on the four Nevanlinna functions of two variables associated with the moment problem.
   The talk is based on joint work with Ryszard Szwarc, Poland.
5. 13 February 2026 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
   Kleshchev Multipartitions and $q$-Appell Functions
   Ae Ja Yee
   Pennsylvania State University
   Abstract: In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koike algebras $\mathcal{H}_{\mathbb{C},v;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (when the parameters are roots of unity and $v \neq 1$) are labeled by the so-called Kleshchev multipartitions. This together with Ariki's categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl--Kac character formula. In this talk, I will revisit this generating function for $v=Q_1=\cdots=Q_{a}=-1, Q_{a+1} =\cdots =Q_{m}=1$. This case is particularly interesting, for the corresponding Kleshchev multipartitions have a close connection to the Rogers--Ramanujan identities. I will discuss an analytic proof of this generating function. While studying enumerative aspects of Kleshchev bipartitions, i.e., the $m=2$ case, two double sum evaluations were found. The second objective of this talk is to give a connection of these two identities to some $q$-Appell functions. I will also discuss a generalization of each evaluation.
   This talk is based on joint work with S. Chern, Z. Li, D. Stanton, T. Xue, and R. Li, S. Seo, D. Stanton.