You are here

Home » Research » Seminars & Colloquia

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.

  • 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).
  • 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.
  • 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.
  • 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.

Fall 2025

For the Fall 2025 semester we will meet virtually on zoom at 11:00 am on Fridays. For more information contact Mourad Ismail or Xiang-Sheng Wang.

  • 5 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
    Al-Salam–Chihara polynomials and limits of random Motzkin paths
    Alexey Kuznetsov
    York University, Canada
    Abstract: The central theme for this talk is the interplay between probability and analysis. We begin by discussing Motzkin paths with general weights and their connection with orthogonal polynomials. Next, we examine the limiting behavior of the initial and final segments of a random Motzkin path, as well as the macroscopic limits of the resulting processes. These results rely on the behavior of the Al-Salam–Chihara polynomials near the right endpoint of their orthogonality interval, along with the limiting properties of the q-Pochhammer and q-Gamma functions. The significance of these findings lies in the fact that these limiting processes also arise in the description of the stationary measure for the KPZ equation on the half-line and of the conjectural stationary measure of the hypothetical KPZ fixed point on the half-line.
  • 12 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
    Nonlinear extension of the J-matrix method of scattering
    Abdulaziz Alhaidari
    Saudi Center for Theoretical Physics
    Jeddah, Saudi Arabia
    Abstract: The J-matrix method was developed in the mid 1970s by a group of physicists at Harvard University that included: Heller, Yamani, Reinhardt, et. al. The method describes quantum scattering due to short-range linear interaction potentials. It compares favorably to other well established scattering methods with enhanced accuracy and convergence. It was applied successfully in atomic, molecular, and nuclear physics. The method was turned into a rigorous mathematical technique by Ismail, Koelink, et. al. Here, we introduce an extension of the method to nonlinear self-interaction potentials. The extension relies predominantly on the linearization of products of orthogonal polynomials.
  • 19 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
    Hypergeometric Functions and Modular Forms
    Ling Long
    Louisiana State University
    Abstract: The theories of hypergeometric functions and modular forms are highly intertwined. In this talk, we will give an overview of the theories leading to an explicit “Hypergeometric-Modularity” method for associating a modular form to a given hypergeometric datum. It is based on joint papers with Michael Allen, Brian Grove and Fang-Ting Tu, as well as recent papers by Esme Rosen.
  • 26 September 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
    Finite asymptotic expansion for the energy of greedy sequences on the unit circle, and density of limit points
    Abey López-García
    University of Central Florida
    Abstract: We consider sequences on the unit circle obtained by a greedy algorithm which optimizes at each step the Riesz potential generated by previously selected points. Using an asymptotic series expansion due to Brauchart, Hardin, and Saff for the Riesz energy of equally spaced points on the unit circle, we give a finite asymptotic expansion for the energy of the first N points of a greedy sequence. We also show that for all values of the Riesz parameter s>-1, the normalized energy has limit points that fill out an interval, as it was expected from numerical experiments. The density of the limit points follows from the continuity of certain extensions of arithmetic functions defined on the interval [1/2, 1]. This talk is based on joint work with Erwin Mina-Diaz.
  • 3 October 2025 (on Zoom: https://ullafayette.zoom.us/j/2022002220)
    Riesz Interpolation Formula for Askey-Wilson Operators and Beyond
    Dr. Xin Li
    University of Central Florida
    Abstract: This talk presents recent work with Rajitha Ranasinghe and Seok-Young Chung, tracing a path from classical interpolation formulas to modern generalizations in $q$-calculus. We begin with the Riesz Interpolation Formula for the derivatives of trigonometric functions and its generalization to functions of exponential type by Boas. We then detail our extension of this classical result to the Askey-Wilson operator, an important difference operator in $q$-orthogonal polynomials. A compelling consequence of this new formula is its equivalence to the classical Sampling Theorem, a connection revealed by the freedom afforded by the parameter $q$. The talk will conclude with a preview of ongoing work extending these ideas to a broad class of difference operators.