Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Pade interpolants, from which convergence follows.Comment: 42 pages, 3 figure

Parameter and q asymptotics of Lq-norms of hypergeometric orthogonal polynomials

The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic Lq-norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér-Rao, the Fisher-Shannon and LMC measures of complexity. In this work, we examine review and solve the q-asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted Lq-norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity-like properties of the highly excited Rydberg and high-dimensional pseudo-classical states of harmonic (oscillator-like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecu- lar systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator-like.].The work of J.S. Dehesa has been partially supported by the grant I+D+i of Junta de Andalucia with ref. P20-00082, and the grant PID2020-113390GB-I00 of the Agencia Estatal de Investigación (Spain) and the European Regional Development Fund (FEDER). The work of N. Sobrino has been partially supported by the grant IT1249-19 of Basque Government and UPV/EHU

A quantization of the harmonic analysis on the infinite-dimensional unitary group

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞). A way to describe those measures is to convert them into determinantal point processes on the real line; it turned out that their correlation kernels are computable in explicit form - they admit a closed expression in terms of the Gauss hypergeometric function F12 (Borodin and Olshanski [8] ).In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials - the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function ϕ12.A crucial novel ingredient of our approach is an extended version G of the Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin [17]). Keywords: Noncommutative harmonic analysis; Gelfand–Tsetlin graph; Determinantal measure

Exceptional Laguerre polynomials

The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way, and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.Comment: To appear in Studies in Applied Mathematic

Associated Legendre Functions and Spherical Harmonics of Fractional Degree and Order

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions, belong to types classified by Schwarz, with dihedral, tetrahedral, or octahedral monodromy. The dihedral Legendre functions are expressed in terms of Jacobi polynomials. For the last two monodromy types, an underlying `octahedral' polynomial, indexed by the degree and order and having a non-classical kind of orthogonality, is identified, and recurrences for it are worked out. It is a (generalized) Heun polynomial, not a hypergeometric one. For each of these families of algebraic associated Legendre functions, a representation of the rank-2 Lie algebra so(5,C) is generated by the ladder operators that shift the degree and order of the corresponding solid harmonics. All such representations of so(5,C) are shown to have a common value for each of its two Casimir invariants. The Dirac singleton representations of so(3,2) are included.Comment: 44 pages, final version, to appear in Constructive Approximatio