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Register a new userPearson Equations for Discrete Orthogonal Polynomials: II. Generalized Charlier, Meixner and Hahn of type I cases
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The Cholesky factorization of the moment matrix is considered for the generalized Charlier, generalized Meixner and generalized Hahn of type I discrete orthogonal polynomials. For the generalized Charlier we present an alternative derivation of the Laguerre-Freud relations found by Smet and Van Assche. Third order and second order order nonlinear ordinary differential equations are found for the recursion coefficient $gamma_n$. Laguerre-Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche. Finally, the generalized Hahn of type I discrete orthogonal polynomials are studied as well, and Laguerre-Freud equations are found and the differences with the equations found by Dominici and by Filipuk and Van Assche are given.
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The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semi-infinite matrix that models the shifts by $pm 1$ in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous Toda for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It also shown that the Kadomtesev-Petvishvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case the deformation do not satisfy a Pearson equation.
In this contribution we consider the sequence ${Q_{n}^{lambda}}_{ngeq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences begin{equation*} langle p,qrangle _{lambda}=int_{0}^{infty}pleft(xright) qleft(xright) dpsi ^{(a)}(x)+lambda ,Delta p(c)Delta q(c), end{equation*} where $lambda in mathbb{R}_{+}$, $Delta $ denotes the forward difference operator defined by $Delta fleft(xright) =fleft(x+1right) -fleft(xright) $, $psi ^{(a)}$ with $a>0$ is the well known Poisson distribution of probability theory% begin{equation*} dpsi ^{(a)}(x)=frac{e^{-a}a^{x}}{x!}quad text{at}x=0,1,2,ldots, end{equation*}% and $cin mathbb{R}$ is such that $psi ^{(a)}$ has no points of increase in the interval $(c,c+1)$. We derive its corresponding hypergeometric representation. The ladder operators and two differe
An algebra denoted $mmathfrak{H}$ with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on $mmathfrak{H}$. Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of $mmathfrak{H}$ in $mathcal{U}(mathfrak{sl}_2)$ is presented. A Pade approximation table for the binomial function is obtained as a by-product.
The present paper is about Bernstein-type estimates for Jacobi polynomials and their applications to various branches in mathematics. This is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of Schrodinger equations whose Hamiltonian is given by the generalized Laguerre operator. More precisely, we show that dispersive estimates for the Schrodinger equation associated with the generalized Laguerre operator are connected with Bernstein-type inequalities for Jacobi polynomials. We use known uniform estimates for Jacobi polynomials to establish some new dispersive estimates. In turn, the optimal dispersive decay estimates lead to new Bernstein-type inequalities.
Over the last decade it has become clear that discrete Painleve equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painleve equation and determining its type according to Sakais classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakais geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painleve equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painleve equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painleve-V equation. We also provide an explicit change of variables transforming this equation to the standard form.
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