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1. Local Asymptotics for Orthonormal Polynomials on the Unit Circle Via Universality

   Lubinsky, D. ( January 2020 , Journal d'Analyse Mathematique (it won't recognize the journal))

   null (Ed.)

   Letbe a positive measure on the unit circle that is regularin the sense of Stahl, Totik, and Ullmann. Assume that in some subarcJ,is absolutely continuous, while0is positive and continuous. Letf'ngbe the orthonormal polynomials for. We show that for appropriaten2J,'n(n(1+zn))'n(n)n1is a normal family in compact subsets ofC. Usinguniversality limits, we show that limits of subsequences have the formez+C(ez 

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2. The effect of adding mass points on bounds for orthogonal polynomials

   Lubinsky, D. ( January 2021 , Dolomites Research Notes on Approximation)

   Let ν be a positive measure supported on [-1,1], with infinitely many points in its support. Let {p_{n}(ν,x)}_{n≥0} be its sequence of orthonormal polynomials. Suppose we add masspoints at ±1, giving a new measure μ=ν+Mδ₁+Nδ₋₁. How much larger can |p_{n}(μ,0)| be than |p_{n}(ν,0)|? We study this question for symmetric measures, and give more precise results for ultraspherical weights. Under quite general conditions, such as ν lying in the Nevai class, it turns out that the growth is no more than 1+o(1) as n→∞. 

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3. Pointwise Asymptotics for Orthonormal Polynomials at the Endpoints of the Interval Via Universality

   Lubinsky, Doron ( January 2019 , International mathematics research notices)

   We show that universality limits and other bounds imply pointwise asymptotics for orthonormal polynomials at the endpoints of the interval of orthonormality. As a consequence, we show that if μ is a regular measure supported on [−1, 1], and in a neighborhood of 1, μ is absolutely continuous, while for some α > −1, μ (t) = h (t)(1 − t) α, where h (t) → 1 as t → 1−, then the corresponding orthonormal polynomials {pn} satisfy the asymptotic limn→∞pn1 − z22n2pn (1) = J∗α (z)J∗α (0) uniformly in compact subsets of the plane. Here J∗α (z) = Jα (z) /zα is the normalized Bessel function of order α. These are by far the most general conditions for such endpoint asymptotics 

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4. Stable Evaluation of 3D Zernike Moments for Surface Meshes

   https://doi.org/10.3390/a15110406  

   Houdayer, Jérôme ; Koehl, Patrice ( November 2022 , Algorithms)

   The 3D Zernike polynomials form an orthonormal basis of the unit ball. The associated 3D Zernike moments have been successfully applied for 3D shape recognition; they are popular in structural biology for comparing protein structures and properties. Many algorithms have been proposed for computing those moments, starting from a voxel-based representation or from a surface based geometric mesh of the shape. As the order of the 3D Zernike moments increases, however, those algorithms suffer from decrease in computational efficiency and more importantly from numerical accuracy. In this paper, new algorithms are proposed to compute the 3D Zernike moments of a homogeneous shape defined by an unstructured triangulation of its surface that remove those numerical inaccuracies. These algorithms rely on the analytical integration of the moments on tetrahedra defined by the surface triangles and a central point and on a set of novel recurrent relationships between the corresponding integrals. The mathematical basis and implementation details of the algorithms are presented and their numerical stability is evaluated. 

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5. Equidistribution and partition polynomials

   Folsom, Amanda ; Males, Joshua ; Rolen, Larry ( August 2023 , The Ramanujan journal)

   Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including spt-crank, overpartition pairs, and t-core partitions. As corollaries, we obtain new proofs of various Ramanujan-type congruences for associated partition functions. Moreover, using results of Erdös and Turán, we establish the equidistribution of roots of partition polynomials on the unit circle including those for the rank, crank, spt, and unimodal sequences. Our results complement earlier work on this topic by Stanley, Boyer-Goh, and others. We explain how our methods may be used to establish similar results for other partition polynomials of interest, and offer many related open questions and examples. 

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