On the second-order holonomic equation for Sobolev-type orthogonal polynomials

Abstract

It is presented a general approach to the study of orthogonal polynomials related to Sobolev inner products which are defined in terms of divided-difference operators having the fundamental property of leaving a polynomial of degree n − 1 when applied to a polynomial of degree n. This paper gives analytic properties for the orthogonal polynomials, including the second order holonomic difference equation satisfied by them.

On Second Order q-Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

“…It is a direct consequence of the connection Formulas ( 17)-( 20), the three-term recurrence relation ( 9) satisfied by {U (a) n (x; q)} n≥0 , and the structure relation (10). To be more precise, applying the q-derivative operator D q to (17) for = −1, 1, together with the property (5) yields…”

“…n−1 (x; q), leading to (17). At this point, we provide another relation between the two families of polynomials, which will be applied in Theorem 1.…”

“…At this point, we provide another relation between the two families of polynomials, which will be applied in Theorem 1. More precisely, from (17) and the recurrence relation (9) we have that…”

“…For n = 0, a trivial verification shows that (21) yields U (a) 0 (x; q) = 1. For n ≥ 1, combining (8) with (17) and the relations…”

“…There, the authors generalize the action of an arbitrary number of q-derivatives for general orthogonality measures, using the same techniques as for example in [16], and also in the present paper. It is also worth mentioning the nice variation considering special non-uniform lattices (snul), instead of uniform or q-lattices, studied in the recent work [17].…”