Abstract
This contribution deals with the sequence {Un(a)(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 Un(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.
Highlights
The Al-Salam–Carlitz I and II orthogonal polynomials of degree n, usually denoted in the literature as Un ( x; q) and Vn ( x; q) respectively, are two systems of one parameter q-hypergeometric polynomials introduced in 1965 by W
We provide four different versions of the second order q-difference equations satisfied by the family of orthogonal polynomials under consideration
We provide up to four different versions of the second order ( a) q-difference equations satisfied by the monic polynomials {Un ( x; q, j)}n≥0, orthogonal with respect to a Sobolev-type inner product associated to the Al-Salam–Carlitz I orthogonal polynomials
Summary
The Al-Salam–Carlitz I and II orthogonal polynomials of degree n, usually denoted in the literature as Un ( x; q) and Vn ( x; q) respectively, are two systems of one parameter q-hypergeometric polynomials introduced in 1965 by W. To the best of our knowledge, an arbitrary number of q-derivatives acting at the same time on the two boundaries of a bounded orthogonality interval, has never been previously considered in the literature, and the present work is intended to be a first step in this direction This reveals some small differences of the corresponding polynomial sequences, for example, related with the parity of the polynomials, with respect to what happens considering only one mass point (as in [15]), and that we have under study. A final section on conclusions and future research problems is included
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