On Factorization and Solutions ofq-difference Equations  Satisfied by some Classes of Orthogonal Polynomials

Abstract

We derive and factorize the fourth-order q-difference equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coefficients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified q-classical orthogonal polynomials. Moreover we find a basis of four linearly independent solutions of these fourth-order q-difference equations and express the modified families in terms of the starting ones.

On difference equations for orthogonal polynomials on nonuniform lattices1

“…The four linearly independent solutions given in the previous corollary are still valid but one should keep in mind that P r and Q r in this cases represent the two linearly independent solutions of f ðxðsÞÞF x yðxðsÞÞ þ c ðxðsÞÞM x yðxðsÞÞ þ l r yðxðsÞÞ ¼ 0 for the real number r [31]. This extension can be deduced following the method used for the rth associated classical orthogonal polynomials of a continuous variable (see [12], Theorem 8).…”

“…Other modifications of the three-term recurrence relation which lead to relations of the type (12) are the co-recursive and the generalized co-recursive orthogonal polynomials; the co-recursive associated and the generalized co-recursive associated orthogonal polynomials; the co-dilated and the generalized co-dilated orthogonal polynomials; the co-modified and the generalized co-modified orthogonal polynomials. Information about these families of orthogonal polynomials as well as the relations of type (12) they satisfy can be found in Refs.…”

“…obtained from the fact that P nþr , Q nþr and P ðrÞ n satisfy the second-order recurrence relation (see Theorem 9 and also [12]) to get P n ðxðsÞÞ ¼ S 3 ðs; n; r; kÞ þ g 21 0 S 2 ðs; n; r; kÞ:…”

“…where P ð1Þ nþr21 is defined in (13) and (14), r and k are nonnegative integers, I n,r,k (x) and J n,r,k (x) are polynomials in the variable x with the property that they do not depend on n for n $ k, i.e. : I n;r;k U I r;k ; J n;r;k U J r;k -0; n $ k:…”

Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights

“…Similar to the algorithm used in [5] we have the following theorem: Theorem: The q-normal equations for …”