On second order q-difference equations satisfied by Al-Salam-Carlitz I-Sobolev type polynomials of higher order

TL;DR

This paper studies higher-order Sobolev-type polynomials related to Al-Salam-Carlitz I polynomials, deriving second-order q-difference equations, connection formulas, and a novel three-term recurrence with rational coefficients.

Contribution

It introduces new Sobolev-type polynomials with q-derivatives, establishes their second-order q-difference equations, and presents a novel three-term recurrence with rational coefficients.

Findings

Derived second-order q-difference equations for the polynomials.

Established connection formulas and ladder operators.

Presented a new three-term recurrence with rational coefficients.

Abstract

This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order $q$-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by $\mathbb{U}_{n}^{(a)}(x;q,j)$, which paves the way to establish an appealing generalization of the so-called $J$-fractions to the framework of Sobolev-type orthogonality.