Semi-classical Orthogonal Polynomials Of Class Research Articles

The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their Extensions

We study a large class of orthogonal polynomials, containing the semi-classical symmetric orthogonal polynomials of class two. We show difference equations for the recurrence coefficients of the orthogonal polynomials as well as for related quantities. Some of these recurrences are identified with discrete Painleve equations.

Discrete semi-classical orthogonal polynomials of class one on quadratic lattices

ABSTRACTWe study orthogonal polynomials on quadratic lattices with respect to Stieltjes functions, S, that satisfy a difference equation where A is a polynomial of degree less or equal than 3 and C is a polynomial of degree greater or equal than 1 and less or equal than 2. We show systems of difference equations for the orthogonal polynomials that arise from the so-called compatibility conditions. Some closed formulae for the recurrence relation coefficients are obtained.

Cubic decomposition of a family of semiclassical orthogonal polynomials of class two

ABSTRACTWe deal with a family of semiclassical orthogonal polynomial sequences of class two having the cubic decomposition , . Only four monic orthogonal polynomial sequences (MOPS) appear in which their recurrence coefficients are explicitly given.

A family of discrete semi-classical orthogonal polynomials of class one

In this work, we solve the system of Laguerre–Freud equations for the recurrence coefficients $$\beta _n$$ , $$\gamma _{n+1} , n \ge 0$$ of the $$D_{w}$$ -semi-classical orthogonal polynomials sequences of class one in the case when $$\beta _{0}=-t_{0}$$ , $$\beta _{n+1}=t_{n}-t_{n+1}$$ and $$\gamma _{n+1}=-t_{n}^{2}$$ with $$t_{n}\ne 0\;n\ge 0$$ , where $$D_w$$ is the divided difference operator. There are essentially four canonical families.

On semiclassical orthogonal polynomials via polynomial mappings

We consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class. We prove that this class is stable under polynomial transformations. Several consequences of this fact are deduced. As an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2, recovering and extending some results proved recently for class 1, and producing new examples of semiclassical orthogonal polynomials of class 2. In particular, we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal.

Discrete semiclassical orthogonal polynomials of class one

We study the discrete semiclassical orthogonal polynomials of class s=1. By considering all possible solutions of the Pearson equation, we obtain five canonical families. We also consider limit relations between these and other families of orthogonal polynomials.

Q-extension of some symmetrical and semi-classical orthogonal polynomials of class one

We study in detail a q-extension of a symmetrical form (functional) of class one. We show that it is symmetrical and Hq-semi-classical of class one. The moments and a discrete representation are given.

The symmetric D ω -semi-classical orthogonal polynomials of class one

We give the system of Laguerre–Freud equations associated with the D ω -semi-classical functionals of class one, where D ω is the divided difference operator. This system is solved in the symmetric case. There are essentially two canonical cases. The corresponding integral representations are given.

Some symmetric semi-classical orthogonal polynomials of class 0≤s≤5

The system fulfilled by symmetric semi-classical orthogonal polynomial sequences of class s = 2 is not linear and has not been sorted out. Alaya and Bouras [Alaya, J. and Bouras B., Symmetric semi-classical forms of class s = 2. Integral Transforms and Special Functions, in press.] used Hankel determinants to give only one case of these sequences. In this paper, we present a method which investigates not only this previous case, but also gives new ones of class s≥2.

Discrete semi-classical orthogonal polynomials: generalized Charlier

Generalized Charlier polynomials are introduced as semi-classical orthogonal polynomials of class 1 with one parameter. Main characteristic data are established from the Laguerre–Freud equations generating the coefficients of the recurrence relation satisfied by the polynomials.

Laguerre-Freud equations for the recurrence coefficients of Dω semi-classical orthogonal polynomials of class one

The Laguerre-Freud equations giving the recurrence coefficients β n , γ n of orthogonal polynomials with respect to a D ω semi-classical linear form are derived. D ω is the difference operator. The limit when ω → 0 are also investigated recovering known results. Applications to generalized Meixner polynomials of class one are also treated.