Abstract
The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.
Highlights
Let μ(x) denote a nondecreasing real-valued, bounded function with a finite or an infinite number of points of increase in the interval, [a, b]
̄ The building of a bridge between the theory of Magnus based mainly on the Riccati Equation satisfied by the formal Stieltjes function [17,18] and the theory of orthogonal polynomials on nonuniform lattices based on the functional approach
We provide three applications for the basis, Fk : The first gives a formal expansion of a given function in the basis, Fn, the second provides a new representation of the formal Stieltjes series in terms of the basis, Fk, while the third gives a representation of the basic exponential and trigonometric functions in terms of the basis, Fk, after solving explicitly a second-order divided-difference Equation in terms of Fn
Summary
Let μ(x) denote a nondecreasing real-valued, bounded function with a finite or an infinite number of points of increase in the interval, [a, b]. Classical orthogonal polynomials of a continuous variable, (Pn )n , can be characterized by the distributional differential Equation (usually called Pearson-type Equation) satisfied by their corresponding regular functional, L, [4]: D(φL) = ψL (5). Semi-classical orthogonal polynomials of a continuous variable [4] are defined as those for which the corresponding linear functional, L, satisfies a Pearson-type Equation (5), but with a more relaxed condition on the polynomials, φ and ψ, namely, deg(ψ) ≥ 1. They fulfill important relations—called product and quotient rules—which read, taking into account the shift (compared to the definition in [14]), in the definition of the above defined companion operators as: Theorem 2. [14]
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