Linear graininess time scales and ladder operators of orthogonal polynomials (Q370799)

This is a very nice and interesting paper, which is devoted to the study of orthogonal polynomials (OPs) on linear graininess time scales (LGTSs). The paper consists of seven sections. After the introduction, a brief survey of the foundations of the time scale calculus is given in Section 2. The concept of LGTSs is introduced in Section 3. It is shown (see Theorem~1) that these time scales admit essentially only four types (reals, the \(h\)-equidistant grid, the \(q\)-grid, and the mixed \((q,h)\)-grid). Their properties are further studied with a special attention paid to corresponding integral expressions. In Section 4, OPs on LGTSs are defined and their basic properties are presented. In Section 5, ladder operators are investigated for OPs on LGTSs and a detailed comparison with the existing literature is given. In Section 6, a relation between the coefficients of the lowering operator and of the three-term recurrence relation for OPs on LGTSs is established (see Theorem 3). Finally, functions of the second kind associated with OPs on LGTSs are introduced and the corresponding lowering dynamic equation is derived (see Theorem 4).