Orthogonal Polynomials Associated with a Δ-Sobolev Inner Product

Some remarks on a paper by L. Carlitz

“…An interesting property of these polynomials is that they do not form an orthogonal set with respect to the standard inner-products, but they are orthogonal with respect to some new inner-products involving differential or difference operators. These classes of polynomials (called Sobolev orthogonal polynomials) have been the object of much attention in the last years (see [4], [11], [12], [14], [18], [25], [27], [28], [29], [30], [33], [37], [39], [41], [44], [45], [46], [47], [49], [52]).…”

On the Krall-type discrete polynomials

“…Let us also point out that there are also the so-called discrete (see, e.g., [8]) and qdiscrete Sobolev type orthogonal polynomials associated with the classical discrete and q-classical functionals [23,24]. In both cases the corresponding polynomials can be reduced to the Krall-type one (except for the q-case when the mass is added at zero where a more careful study is needed [23,24]) since the differences ∆f (…”