Recovering the orthogonal polynomials from its specific spectral transformations

In this work, the concept of quasi-type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. Recovering the source orthogonal polynomial involving the linear spectral transformation is provided. This process involves an expression of ratio of kernel polynomials. Special cases of such ratios in terms of certain continued fractions are exhibited.Comment: 21 PAGE

Real roots of hypergeometric polynomials via finite free convolution

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.Comment: 44 pages, 8 table

(M, N)-coherent pairs of order (m, k) and Sobolev orthogonal polynomials

A pair of regular linear functionals (U,V)(U,V) is said to be a (M,N)(M,N)-coherent pair of order (m,k)(m,k) if their corresponding sequences of monic orthogonal polynomials...We thank the referees for the careful revision of the manuscript. Their comments and suggestions have contributed to improve substantially the presentation. The work of F. Marcellán, J. Petronilho, and N.C. Pinzón-Cortés has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grants MTM2012-36732-C03-01 (FM and NCP-C) and MTM2012-36732-C03-02 (JP). The work of J. Petronilho was also supported by the Centro de Matemática da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0324/2011, and the research project PTDC/MAT/098060/2008 (FCT)

Moments of random matrices and hypergeometric orthogonal polynomials

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments \mathbb{E}\Tr X_n^{-s} as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden \textit{et al.}~[F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order $n\to\infty$ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials