The generalized Bochner condition about classical orthogonal polynomials revisited

We bring a new proof for showing that an orthogonal polynomial sequence is classical if and only if any of its polynomial fulfils a certain differential equation of order 2k, for some k?1. So, we build those differential equations explicitly. If k=1, we get the Bochner's characterization of classical polynomials. With help of the formal computations made in Mathematica, we explicitly give those differential equations for k=1,2 and 3 for each family of the classical polynomials. Higher order differential equations can be obtained similarly.     

Characterizations of classical orthogonal polynomials on quadratic lattices

This paper is devoted to characterizations of classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal polynomials on quadratic lattices. Moreover a new matrix characterization of classical ortho-gonal polynomials in quadratic lattices is presented. From the Bochner type characterization we derive the three-term recurrence relation coefficients for these polynomials.     

On the Bochner theorem on orthogonal operators

We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation

where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

     

Difference equations for discrete classical multiple orthogonal polynomials

For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r+1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials.     

On differential properties for bivariate orthogonal polynomials

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.      

Second order partial differential equations for gradients of orthogonal polynomials in two variables

In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one.     

Identities involving Narayana polynomials and Catalan numbers

We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.     

On linearly related sequences of derivatives of orthogonal polynomials

We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (P_n)_n and (Q_n)_n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as

for all n=0,1,2,…, where M and N are fixed nonnegative integer numbers, and r_i,n and s_i,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (P_n)_n and (Q_n)_n (resp.). Assuming 0mk, we prove the existence of four polynomials Φ_M+m+i and Ψ_N+k+i, of degrees M+m+i and N+k+i (resp.), such that

the (k−m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k=m, then u and v are connected by a rational modification. If k=m+1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k>m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k−m with polynomial coefficients.     

A construction of real weight functions for certain orthogonal polynomials in two variables

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are

(x²+y)uxx+2xyuxy+y²uyy+gxux+g(y−1)uy=λu,     

Two classes of special functions using Fourier transforms of some finite classes of classical orthogonal polynomials

Some orthogonal polynomial systems are mapped onto each other by the Fourier transform. The best-known example of this type is the Hermite functions, i.e., the Hermite polynomials multiplied by , which are eigenfunctions of the Fourier transform. In this paper, we introduce two new examples of finite systems of this type and obtain their orthogonality relations. We also estimate a complicated integral and propose a conjecture for a further example of finite orthogonal sequences.

     

Complete solution of the electrostatic equilibrium problem for classical weights

In this paper we complete the solution of the electrostatic equilibrium problem for all classical weight functions. In particular we solve this problem for classical generalized Bessel, Jacobi on (0, +∞) and pseudo-Jacobi weights. In addition we present an elementary unified way of dealing with the electrostatic equilibrium problem, which depends on the weights satisfying the Pearson differential equation.     

Around operators not increasing the degree of polynomials

We present a generic operator J defined on the vectorial space of polynomial functions and we address the problem of finding the polynomial sequences that coincide with the (normalized) J-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. We provide examples for a J limited to three terms.     

Unification of Stieltjes‐Calogero type relations for the zeros of classical orthogonal polynomials

The classical orthogonal polynomials (COPs) satisfy a second‐order differential equation of the form σ(x)y^′′+τ(x)y^′+λy = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed‐form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.     

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.