We investigate the distribution of zeros of the little q-Jacobi polynomials and related q-hypergeometric families. We prove that the zeros of these orthogonal polynomials exhibit strong interlacing properties and obey natural monotonicity rules with respect to the parameters. A key tool in our approach is the logarithmic mesh, which quantifies the relative spacing of the positive real zeros and allows us to classify families of polynomials with prescribed interlacing patterns. Our results include new interlacing relations, monotonicity with respect to parameters, and structural decompositions in non-orthogonal regimes. Several classical families of q-hypergeometric polynomials, including q-Bessel and Stieltjes-Wigert polynomials, are treated as limit cases. The methods rely on a combination of classical orthogonality theory and q-difference equations.

We explore a class of meromorphic functions on elliptic curves, termed elliptic orthogonal a-polynomials (a-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on Bertola's construction of orthogonal sections, we study these functions via non-Hermitian orthogonality on the torus, establish their recurrence properties, and derive an analogue of the Christoffel–Darboux formula. We demonstrate that, under real-valued orthogonality conditions, a-EOPs exhibit interlacing and simplicity of zeros similar to orthogonal polynomials on the real line (OPRL). Furthermore, we construct a general correspondence between families of OPRL and elliptic orthogonal functions, including a decomposition into multiple orthogonality relations, and identify new interlacing phenomena induced by rational deformations of the orthogonality weight.

Zernike polynomials are widely used in optics and ophthalmology due to their direct connection to classical optical aberrations. While orthogonal on the unit disk, their application to discrete data or non-circular domains--such as ellipses, annuli, and hexagons--presents challenges in terms of numerical stability and accuracy. In this work, we extend Zernike-like orthogonal functions to these non-standard geometries using diffeomorphic mappings and construct sampling patterns that preserve favorable numerical conditioning. We provide theoretical bounds for the condition numbers of the resulting collocation matrices and validate them through extensive numerical experiments. As a practical application, we demonstrate accurate wavefront interpolation and reconstruction in segmented mirror telescopes composed of hexagonal facets. Our results show that appropriately transferred sampling configurations, especially Optimal Concentric Sampling and Lebesgue points, allow stable high-order interpolation and effective wavefront modeling in complex optical systems. Moreover, the Optimal Concentric Samplings can be computed with an explicit expression, which is a significant advantage in practice.

We address the problem of the weak asymptotic behavior of zeros of families of generalized hypergeometric polynomials as their degree tends to infinity. The main tool is the representation of such polynomials as a finite free convolution of simpler elements; this representation is preserved in the asymptotic regime, so we can formally write the limit zero distribution of these polynomials as a free convolution of explicitly computable measures. We derive a simple expression for the S-transform of the limit distribution, which turns out to be a rational function, and a representation of the Kampé de Fériet polynomials in terms of finite free convolutions. We apply these tools, as well as those from [42], to the study of some well-known families of multiple orthogonal polynomials (Jacobi–Piñeiro and multiple Laguerre of the first and second kinds), obtaining results on their zeros, such as interlacing, monotonicity, and asymptotics.

Given a sequence of polynomials Q_n of degree n with zeros on [-1,1], we consider the triangular table of derivatives Q_n, k(x)=d^k Q_n(x) /d x^k. Under the assumption that the sequence {Q_n} has a weak* limiting zero distribution (an empirical distribution of zeros) given by the arcsine law, we show that as n, k →∞ such that k / n → t ∈[0,1), the zero-counting measure of the polynomials Q_n, k converges to an explicitly given measure μ_t. This measure is the equilibrium measure of [-1,1] of size 1-t in an external field given by two mass points of size t/2 fixed at ± 1. The main goal of this paper is to provide a direct potential theory proof of this fact.

For a monic polynomial Q_n of degree n, let Q_n, k be its k-th derivative normalized to be monic. Under the only assumption that the sequence {Q_n} has a weak* limiting zero distribution (an empirical distribution of zeros) represented by a probability measure μ_0 with compact support in the complex plane, we show that as n, k →∞ such that k / n → t ∈(0,1), the Cauchy transform of the normalized zero-counting measure of the polynomials Q_n, k converges in a neighborhood of infinity to an analytic function, uniquely determined by μ_0 and t, that can be written as the Cauchy transform of a measure μ_t, not necessarily uniquely determined unless μ_0 is supported on the real line. The family of these Cauchy transforms and, when well defined, the corresponding measures μ_t, t ∈(0,1), whose dependence on the parameter t can be interpreted as a flow of the zeros under iterated differentiation, has several interesting connections with the inviscid Burgers equation, the fractional free convolution of μ_0, or a nonlocal diffusion equation governing the density of μ_t on ℝ. We provide an elementary and unified approach that not only recovers, but also explains various phenomena observed in prior works - from Burgers-type PDEs to free probability limits.

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko-Pastur, reversed Marchenko-Pastur, and free beta laws, which has an independent interest within free probability.

Information about the behavior of zeros of classical families of multiple or Hermite-Pad\'e orthogonal polynomials as functions of the intrinsic parameters of the family is scarce. We establish the interlacing properties of the zeros of Angelesco-Jacobi polynomials when one of the three main parameters is increased by 1, extending the work of dos Santos (2017). We also show their monotonicity with respect to (large values) of the parameter representing in the electrostatic model of the zeros the size of the positive charge fixed at the origin, as well as monotonicity with respect to the endpoint of the interval of orthogonality. These results are extended to zeros of multiple Jacobi-Laguerre and Laguerre-Hermite polynomials using asymptotic relations between these families.

Purpose: generate a deep learning model architecture based on hierarchization of information and the use of attention mechanisms in each component for OCT image-based classification of age macular degeneration (AMD). Design: Validation of a deep learning model ViT architecture to detect and classify AMD from OCT images using a hierarchization of the visual information to reduce the number of trainable parameters. Methods: Design and validation of an Spatial-Coherent Vision Transformers (SCViT) model with progressive downsampling of image patches in each iteration before feeding them to the transformer encoder. Study population: The model was applied to 109,309 OCT images from the subset OCTMNIST of the MedMNIST v2 benchmark of medical datasets publicly available at a GitHub repository (from which, 1,080 corresponded to training) grouped into four classes: "Normal", "Drusen", “DME, and "CNV". The dataset was used to train and validate our prediction model. Main Outcome Measures: accuracy scores and the area under the ROC curve. Results: The Spatial-Coherent Vision Transformers (SCViT) model obtanined the best results for f1-score, accuracy, and ROC AUC for 6 out of the 8 datasets compared to purely convolutional architectures, such as EfficientNet, ConxMixer, and MobileNet. Conclusions: An attention-based architecture adapted to classification problems has been tested as an effective tool to recognize local-global dependencies in structural OCT patterns via attention masks analysis and further explore transformers' capabilities in new research fields.

We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as Szegő polynomials. Our results have connections to classical results from algebraic and projective geometry, such as theorems of Poncelet, Darboux, and Kippenhahn; numerical ranges of a class of matrices; and Blaschke products and disk functions. This paper contains new results, some old results presented from a different perspective or with a different proof, and a formal foundation for our analysis. We give a rigorous definition of the Poncelet property, of curves tangent to a family of polygons, and of polygons associated with Poncelet curves. As a result, we are able to clarify some misconceptions that appear in the literature and present counterexamples to some existing assertions along with necessary modifications to their hypotheses to validate them. For instance, we show that curves inscribed in some families of polygons supported on T are not necessarily convex, can have cusps, and can even intersect the unit circle. Two ideas play a unifying role in this work. The first is the utility of OPUC and the second is the advantage of working with tangent coordinates. This latter idea has been previously exploited in the works of B. Mirman, whose contribution we have tried to put in perspective.

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P . The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P . This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

Given a natural number n≥3 and two points a and b in the unit disk 𝔻 in the complex plane, it is known that there exists a unique elliptical disk having a and b as foci that can also be realized as the intersection of a collection of convex cyclic n-gons whose vertices fill the whole unit circle 𝕋. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of n. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.

In a recent paper (Martínez-Finkelshtein et al. in Proc Am Math Soc 147:2625–2640, 2019) some interesting results were obtained concerning complementary Romanovski–Routh polynomials, a class of orthogonal polynomials on the unit circle and extended regular Coulomb wave functions. The class of orthogonal polynomials here are generalization of the class of circular Jacobi polynomials. In the present paper, in addition to looking at some further properties of the complementary Romanovski–Routh polynomials and associated orthogonal polynomials on the unit circle, behaviour of the zeros of these extended Coulomb wave functions are also studied.

There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the theory of OPUC.

We consider the type I multiple orthogonal polynomials (MOPs) $(A_{n,m}, B_{n,m})$ and type II MOPs $P_{n,m}$, satisfying non-hermitian orthogonality with respect to the weight $e^{-z^3}$ on two unbounded contours on $\mathbb C$. Under the assumption that $$ n,m \to \infty, \quad \frac{n}{n+m}\to \alpha \in (0, 1) $$ we find the detailed asymptotics of the MOPs, and describe the phase transitions of this limit behavior as a function of $\alpha$. This description is given in terms of vector critical measures, which are saddle points of an energy functional comprising both attracting and repelling forces. These critical measures are characterized by a cubic equation (spectral curve), and their components $\mu_j$ live on trajectories of a canonical quadratic differential $\varpi$ on the Riemann surface of this equation, which was object of study in our previous paper [Adv. Math. 302 (2016), 1137--1232]. The asymptotic zero distribution of the polynomials $A_{n,m}$ and $P_{n,m}$ are given by appropriate combinations of the components of the vector critical measure. However, in the case of the zeros of $B_{n,m}$ the behavior is totally different, and can be described in terms of the balayage of $\mu_2 - \mu_3$ onto certain curves on the plane. These curves are constructed with the aid of $\varpi$, and their topology has three very distinct characters, depending on the value of $\alpha$, and are obtained from the critical graph of $\varpi$. Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the analysis of a $3\times 3$ Riemann-Hilbert problem characterizing the MOPs. We illustrate our findings with results of several numerical experiments, and formulate some conjectures and empirical observations based on these experiments.

We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-dimensional Schrodinger equation and also the wave functions of quarks. One of the main results of this paper is to show how the CRR-polynomials are related to a special class of orthogonal polynomials on the unit circle. As another main result, we have established their connection to a class of functions which are related to a subfamily of Whittaker functions that includes those associated with the Bessel functions and the regular Coulomb wave functions. An electrostatic interpretation for the zeros of CRR-polynomials is also considered.

Close-form expression for the Strehl ratio calculated in the spatial frequency domain of the optical transfer function (SOTF) is considered for the case of time-varying dynamic optical system that has circular symmetry. Specifically, closed-form expressions for the temporally averaged SOTF are considered, which can be easily evaluated numerically (what we call a semi-analytic solution). As for the case of a static wavefront, described in Part 1 of this work, it is shown that the proposed methods are computationally more efficient than the commonly used approach based on the discrete Fourier transform.

Close-form expression for the Strehl ratio calculated in the spatial frequency domain of the optical transfer function (SOTF) is considered for the case of an optical system that has circular symmetry. First, it is proved that the SOTF for the aberration-free diffraction limited optical system is equal to one. Further, a semi-analytic solution for the SOTF for a system described by the second (defocus) and the fourth (spherical) order aberrations is provided. It is shown that the proposed semi-analytical solution is of an order of a magnitude computationally more efficient than the commonly used approach based on the discrete Fourier transformation.

Given a nontrivial Borel measure mu on the unit circle T, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by two sequences of real coefficients, {c(n)} and {d(n)}, where {d(n)} is additionally a positive chain sequence. Coefficients (c(n), d(n)) provide a parametrization of a family of measures related to mu by addition of a mass point at z = 1. In this paper we estimate the location of the extreme zeros (those closest to z = 1) of the para-orthogonal polynomials from the (c(n), d(n))-parametrization of the measure, and use this information to establish sufficient conditions for the existence of a gap in the support of mu at z = 1. These results are easily reformulated in order to find gaps in the support of mu at any other z epsilon T. We provide also some examples showing that the bounds are tight and illustrate their computational applications.

Given a nontrivial positive measure it on the unit circle T, the associated Christoffel-Darboux kernels are K-n(z, w; mu) = Sigma(n)(k=0)<(phi(k)(w;mu))over bar>phi(k)(z; mu), n > 0, where phi(k)(.; mu) are the orthonormal polynomials with respect to the measure mu. Let the positive measure nu on the unit circle be given by d nu(z) = vertical bar G(2m) (z)vertical bar d mu(z), where G(2m) is a conjugate reciprocal polynomial of exact degree 2m. We establish a determinantal formula expressing {K-n(z, w; nu)}n >= 0 directly in terms of {K-n(z, w; mu)}n >= 0. Furthermore, we consider the special case of w = 1; it is known that appropriately normalized polynomials K-n(z, 1; mu) satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters {c(n)(mu)}(n=1)(infinity) and {g(n)(mu)(n=1)(infinity), with 0 < g(n) < 1 for n >= 1. The double sequence {c(n)(mu), g(n)(mu))}(n=1)(infinity) characterizes the measure mu. A natural question about the relation between the parameters c(n) (mu), g(n)(mu), associated with mu, and the sequences c(n)(nu), g(n)(nu), corresponding to nu, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a measure for which the Christoffel-Darboux kernels, with w = 1, are given by basic hypergeometric polynomials and a measure for which the orthogonal polynomials and the Christoffel-Darboux kernels, again with w = 1, are given by hypergeometric polynomials. (C) 2018 Elsevier Inc. All rights reserved.