Analysis Seminar

In solving q-difference equations, and in the definition of q-special functions, we encounter formal power series in which the nth coefficient is of size q^{-(n choose 2)} with q ∈ (0,1) fixed. To make sense of these formal series, a q-Borel-Laplace resummation is required. There are three candidates for the q-Laplace transform, resulting in three different resummations. Surprisingly, the differences between these resummations have hardly been discussed in the literature. Our main result provides explicit formulas for these q-exponentially small differences. We also give simple Mellin–Barnes integral representations for all the basic hypergeometric _rφ_s functions. As the main application, we introduce three resummations for the ₂φ₀ function which can be seen as q versions of the Kummer U function. We derive many of their properties, including interesting integral and sum representations, connection formulas, and error bounds.

In their study of the asymptotic iteration method (AIM), Ismail and Saad posed the problem of finding an explicit formula for powers of a matrix differential operator associated with the AIM sequences. We address this problem in the more general setting of matrix differential operators of the form L = D + M(x), where D = d/dx and M(x) is an arbitrary square matrix whose entries are functions of x. An explicit expansion for Lⁿ is obtained in terms of recursively generated coefficient matrices. A complementary representation is developed using an invertible matrix satisfying a first-order matrix differential equation.

For the matrix arising from AIM, the coefficient matrices are identified explicitly with the AIM sequences. This yields the desired operator-power formula, finite-sum representations for the AIM sequences, and the associated termination quantity. Differentiation acts only on the original coefficient functions, avoiding the repeated differentiation inherent in the classical AIM recurrence.

For second-order linear differential equations, whenever a polynomial solution exists, the first stabilized AIM ratio determines the polynomial up to normalization and coincides with its negative logarithmic derivative. Consequently, the zeros of the polynomial are encoded as the poles of this rational function. The framework also leads naturally to finite-term recurrences for AIM sequences associated with classical hypergeometric-type and Heun-type equations. Work in progress with M. E. H. Ismail.

In this talk, we consider the evaluation of definite integrals involving products of modified Bessel functions.

In the first part of the talk, we see how such integrals arise in the calculation of return probabilities in the simple symmetric random walk on the d-dimensional lattice ℤ^d. These probabilities are sometimes referred to as Pólya's random walk constants in recognition of Pólya's beautiful result that the simple symmetric random walk is recurrent in dimensions 1 and 2 but transient in dimension d ≥ 3. We obtain a general formula for the return probability in dimension d ≥ 3 in terms of the Lauricella function of type C, and in dimension 4 we obtain simpler expressions in terms of the generalized hypergeometric function, complementing the known closed-form formulas in dimension 3.

In the second part of the talk, we focus on the evaluation of definite integrals involving the product of four modified Bessel functions and a power function. We provide general formulas expressed in terms of the Meijer G-function, generalized hypergeometric and Lauricella functions, and study a number of special cases in which the integrals can be evaluated in terms of simpler special functions or indeed take an elementary form. As a consequence, we deduce some new formulas for definite integrals of products of four Airy functions.

Part of this talk is joint work with Saralees Nadarajah and Tibor Pogany.

Holomorphic modular forms on congruence subgroups are objects which can be defined analytically and whose Fourier coefficients have deep arithmetic information. These Fourier coefficients obey certain recurrence relations. In this talk, we discuss how these recurrence relations can be extended p-adically for meromorphic modular forms using the geometry of certain vector bundles on modular curves.

Zhedanov associated an algebra defined by generators and relations with the Askey-Wilson polynomials. The notion of nonsymmetric special functions associated with root systems was developed in successive work by Dunkl, Heckman, Cherednik and Sahi. With such functions is associated a Double Affine Hecke Algebra (DAHA), again defined by generators and relations. In the rank one case we have the nonsymmetric Askey-Wilson polynomials and its Askey-Wilson DAHA, depending on a,b,c,d,q just as the polynomials do. Askey-Wilson functions were studied by Suslov, by Ismail & Rahman, and by Koelink & Stokman. Non-symmetric Askey-Wilson functions were next studied by Stokman. They can also be associated with the Askey-Wilson DAHA. In a recent paper [1] together with Marta Mazzocco we discuss Askey-Wilson DAHA automorphisms (affecting both the generators and relations) and how they imply symmetries for the associated special functions. In some cases these are symmetries for Askey-Wilson functions, not for the polynomials. The lecture will present some of these results of [1] and also a glimpse of new work in progress together with Marta Mazzocco and Davide Dal Martello. [1] T.H. Koornwinder and M. Mazzocco, Automorphisms of the DAHA of type Č₁C₁ and non-symmetric Askey-Wilson functions, Indag. Math. (N.S.) 36 (2025), 1795-1829.

I will discuss open problems in (1) integer partitions related to the Rogers-Ramanujan identities, (2) type R_I and type R_II orthogonal polynomials, and, if time permits, positivity questions. This is joint work with many people but mostly with Mourad Ismail.

Ismail and Wilson derived a generating function for Askey–Wilson polynomials which is given by a product of 𝑞-Gauss (Heine) nonterminating basic hypergeometric functions. We provide a generalization of that generating function which contains an extra parameter. A special case gives a closed-form summation formula for a quadruple basic hypergeometric sum. We further present new terminating balanced ₄φ₃ summations that, together with previously derived balanced summations, give rise to 10 𝑞-quadratic two-parameter special values for Askey–Wilson polynomials. We also present new terminating 2-balanced and 3-balanced ₄φ₃ summations. Using the Ismail–Wilson generating function combined with these 𝑞-quadratic special values, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations. Further new identities are obtained by applying Cayley–Orr type expansion formulas. If time permits, we will also show how starting with linear and bilinear generating functions for Askey–Wilson polynomials, we use the previously described 𝑞-quadratic special values to derive 𝑞-quadratic transformation and summation formulas for nonterminating basic hypergeometric series. Joint work with Michael J. Schlosser.

We briefly formulate two challenging questions in number theory: the Gauss circle problem and the problem of the second term in the asymptotics in the Prime Number Theorem, the latter being essentially the Riemann hypothesis. We then pose a general question about possible refinements of the classical Tauberian theorems. Given a sequence of real numbers, it asks for conditions on either the corresponding Dirichlet series or the discrete Laplace transform that would provide information about the second term in the asymptotic behavior of the partial sums.

We then discuss a result that provides a partial answer to a very old open problem of Pólya concerning necessary and sufficient conditions on a kernel for its Fourier transform to be an entire function with only real zeros. This joint work with Yuan Xu uses one of Wiener's Tauberian theorems to produce a density criterion in terms of translates of the kernel in L¹(ℝ). The result yields another necessary and sufficient condition for the Riemann hypothesis, analogous to the Nyman–Beurling and Báez-Duarte density criteria in L²[0, 1] and L²(ℝ).

We will consider the scaling limits of polynomial reproducing kernels for measures on the real line. For many years there has been considerable research to find the weakest assumptions that one can place on a measure that allows one to prove that these rescaled kernels converge to the sinc kernel. Our main result will provide the weakest conditions that have yet been found. In particular, it will demonstrate that one only needs local conditions on the measure. We will also settle a conjecture of Avila, Last, and Simon by showing that convergence holds at almost every point in the essential support of the absolutely continuous part of the measure. This is joint work with Benjamin Eichinger and Milivoje Lukic.

In 2000, Ariki and Mathas showed that the simple modules of the Ariki–Koike algebras H_{ℂ,v;Q₁,…,Qₘ}(G(m, 1, n)) (when the parameters are roots of unity and v ≠ 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 = Q₁ = ⋯ = Q_a = −1, Q_a+1 = ⋯ = Qₘ = 1. This case is particularly interesting because 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 connect 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, R. Li, and S. Seo.

This talk is based on joint work with Ryszard Szwarc. We consider the Jacobi operator (T, D(T)) associated with an indeterminate Hamburger moment problem: the operator in ℓ² defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many nonzero terms. It is well known that it is symmetric with deficiency indices (1, 1).

For a complex number z, let p_z and q_z denote the square-summable sequences corresponding to the orthonormal polynomials pₙ and the polynomials qₙ of the second kind. We determine whether linear combinations of p_u, p_v, q_u, and q_v for u, v ∈ ℂ belong to D(T) or to the domain of the self-adjoint extensions of T in ℓ². The results depend on the four Nevanlinna functions of two variables associated with the moment problem.

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.

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 Román in Pacific J Math 2025.

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× 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, University of Illinois, and Maciej Ulas, Jagiellonian University in Krakow, Poland).

In consideration of the integral transform whose kernel arises as an oscillatory solution of certain second-order linear differential equation, its positivity is investigated on the basis of Sturm’s theory. As applications, positivity criteria are obtained for Hankel transforms as well as trigonometric integrals defined on the positive real line.

We will discuss the asymptotic analysis of matrix-valued orthogonal polynomials (MVOPs) of Jacobi and Hermite type in the large-degree limit. The focus will be on weights consisting of a scalar factor multiplied by a nontrivial matrix part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method of steepest descent, we will analyze the asymptotic behavior of the polynomials in different regions of the complex plane, as well as the corresponding asymptotic behavior of the recurrence coefficients and norms. A key ingredient is the matrix Szegő function, which plays a central role in the construction. The results will be illustrated with explicit examples.

We will introduce matrix-valued orthogonal polynomials (MVOPs) and discuss their main properties. These families arise in representation theory, stochastic processes, and integrable systems, and extend the classical theory of scalar orthogonal polynomials to the matrix setting. We will address reducibility, the distribution of zeros, and the existence of differential operators having the MVOPs as eigenfunctions. Particular emphasis will be placed on the algebraic and analytic tools that make the matrix case both richer and more challenging than the scalar one. Several examples will be presented, including an important class of MVOPs associated with group representations.

Directed last passage percolation is a maximization model defined on a two-dimensional grid. It is naturally related to many other models in statistical physics and engineering. For example, it can be interpreted both as a random growth model and as a serial queuing model. In the large “time” limit, directed last passage percolation is believed, and in some cases proved, to converge to the so-called KPZ fixed point—a universal two-dimensional random field. In this talk, we consider the rare event where the passage time to a certain point is unusually large. In terms of the queuing model, this corresponds to the event that the exit time of a particular customer from a long series of queues is exceptionally large. We condition on the occurrence of such an event, and study its effect on the passage times to other points (i.e., its effect on other customers). Our analysis is based on explicit Fredholm determinant-like formulas for multi-time joint distributions, which were recently obtained by Zhipeng Liu. This talk is based on joint work with Dylan Cordaro and Tejaswi Tripathi.

In this talk, we extend the classical notion of orthogonality to polynomials orthogonal with respect to a q-difference operator. We examine the existence and uniqueness of these orthogonal polynomials and present three sufficient conditions ensuring uniqueness. Several illustrative examples and applications will be provided. In particular, we introduce the polar q-Legendre polynomials and discuss their fundamental structural and analytical properties, including recurrence relations, q-difference equations, asymptotic behavior, and the distribution of zeros.

The distribution of the product of two correlated normal random variables has received much attention in the statistics literature and has numerous applications throughout the applied sciences. However, in practical applications, a key difficulty in working with this distribution is that its probability density function takes a rather complicated form, and exact formulas for other related distributional properties are not known. After reviewing some previous contributions, we present new exact formulas for the density, given by infinite series and integral representations expressed in terms of modified Bessel functions and confluent hypergeometric functions. We then apply these formulas to derive asymptotic expansions for the probability density function, the tail probabilities and the quantile function. We present some numerical results to assess the performance of the asymptotic approximations. This talk is based on joint work with Saralees Nadarajah, Tibor Pogany and Zixin Ye.

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.

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.

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.

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.

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.