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

Abstract

This contribution deals with the sequence {U (a) n (x; q, j)} n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam-Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q ∈ (0, 1). 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 U (a) n (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.