Abstract
In this paper we introduce linear graininess (LG) time scales. We further study orthogonal polynomials (OPs) with the weight function supported on LG time scales and derive the raising and lowering ladder operators by using the time scales calculus. We also derive a second order dynamic equation satisfied by these polynomials. The notion of an LG time scale encompasses the cases of the reals, the h-equidistant grid, the q-grid and, more general, a mixed (q, h)-grid. This allows a unified treatment of the ladder operators theory for classical OPs on these time scales. Moreover we will explain, why exclusively LG time scales provide the right framework for general OP theory.
Highlights
The time scales calculus unifies continuous and discrete analysis
The main objective of this paper is to introduce the linear graininess (LG) time scales and, as an application, show how the theory of ladder operators for orthogonal polynomials (OPs) can be studied by using the time scales calculus
Other articles [6,7] with the same flavor do not fit directly into the framework of time scales, since they are concerned with q-difference operators, whereas the underlying time scale is the set of real numbers R, but not, as would be appropriate from the time scale point of view, the q-grid
Summary
The time scales calculus unifies continuous and discrete analysis. Many analogous results in continuous and discrete analysis can be elegantly proved by using the techniques of time scales calculus. This provides an invaluable insight into many of the investigated problems. Similar proofs in distinct areas of differential and difference calculus can be simplified and be simultaneously written by using the time scales notation. In recent years there have been many studies on dynamic equations (which include differential and difference equations as particular cases) by using the time scales calculus
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