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

This contribution deals with the sequence {𝕌_n^(a)(x;q,j)}_n≥ 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 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.