The birth and death processes are closely related to the orthogonal polynomials. The latter allows determining the stochastic matrix associated with these processes. Let us also note that these processes are stationary Markov processes whose state space is the nonnegative integers. Many authors treated the question of the existing relationship between the birth and death processes and the ortho...

We give a survey of recent generalizations for orthogonal polynomials that were recently obtained. It concerns not only multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, or multipole (orthogonal rational functions) variants of the classical polynomials but also extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations...

Orthogonal polynomials of different kinds as the basic tools play very important role in construction and analysis of quadrature formulas of maximal and nearly maximal algebraic degree of exactness. In this survey paper we give an account on some important connections between orthogonal polynomials and Gaussian quadratures, as well as several types of generalized orthogonal polynomials and corr...

Abstract This manuscript contains a small portion of the algebraic theory orthogonal polynomials developed by Maroni and their applicability to study characterization classical families, namely Hermite, Laguerre, Jacobi, Bessel polynomials. It is presented cyclical proof some most relevant characterizations, particularly those due Al-Salam Chihara, Bochner, Hahn, Maroni, McCarthy. Two apparentl...

It has been recently discovered that exceptional families of SturmLiouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an examp...

From the main equation (ax2 + bx + c)y′′ n (x) + (dx + e)y′ n(x)− n((n− 1)a+ d)yn(x) = 0, n∈ Z+, six finite and infinite classes of orthogonal polynomials can be extracted. In this work, first we have a survey on these classes, particularly on finite classes, and their corresponding rational orthogonal polynomials, which are generated by Mobius transform x = pz−1 + q, p = 0, q ∈ R. Some new int...

An elementary non-technical introduction to group representations and orthogonal polynomials is given. Orthogonality relations for the spherical functions for the rotation groups in Euclidean space (ultraspherical polynomials), and the matrix elements of SU(2) (Jacobi polynomials) are discussed. A general theory for finite groups acting on graphs, giving a finite set of discrete orthogonal poly...