On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

Abstract

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.

Linear Partial Divided-Difference Equation Satisfied by Multivariate Orthogonal Polynomials on Quadratic Lattices

“…as linear combinations of the nine terms R n,m (s + ℓ 1 , t + ℓ 2 ), −1 ≤ ℓ 1 , ℓ 2 ≤ 1, then solving the resulting system of linear equations in terms of the unknowns R n,m (s + ℓ 1 , t + ℓ 2 ), −1 ≤ ℓ 1 , ℓ 2 ≤ 1, and finally replacing the solution into equation (8) and observe that the coefficients obtained are all polynomials in the lattices x(s) and y(t).…”

“…where G n, j are matrices of size (n + 1) × ( j + 1) and G n,n is a nonsingular square matrix of size (n + 1) × (n + 1). Following [8], let us introduce the following bases {F n (x(s))} n∈N and {F n (y(t))} n∈N of monic polynomials in the quadratic lattices x(s) and y(t) defined in (9)…”

On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

“…A. Wilson, M. Ismail [7,8,9,10]; F. Nikiforov, K. Suslov, B. Uvarov, N. M. Atakishiyev [1,2,3,11,12]; G. George, M. Rahman [13]; T. H. Koornwinder [14]; and many other researchers [15,16,17,18,19,20,21,22,23,24,25,26]. On the other hand, many researchers like R.Álvarez-Nodarse, K. L. Cardoso, I.…”

On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

“…Using appropriate bases, computer algebra software has been used to solve divided-difference Equation 8for specific families of classical orthogonal polynomials on a non-uniform lattice. For some special values of the parameter for the specific case of Askey-Wilson polynomials, non-polynomial solution has been recovered together with the polynomial one [14] (see page 15, Equations (62) and (63)). In addition, the operators D x and S x have played a decisive role not only for establishing the functional approach of the characterization theorem of classical orthogonal polynomials on non-uniform lattices, but also for providing algorithmic solution to linear homogeneous divided-difference equations with polynomial coefficients, allowing to solve explicitly [13] the first-order divided-difference equations satisfied by the basic exponential function…”