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 $\{{\mathbb{U}}_n^{(a)}(x;q,j){\}}_{n\ge 0}$ 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.

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