Results in Mathematics

, Volume 64, Issue 1–2, pp 13–35 | Cite as

Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

  • Galina FilipukEmail author
  • Stefan Hilger
Open Access


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.

Mathematics Subject Classification (2000)

26E70 33C47 


Time scales linear graininess time scales ladder operators orthogonal polynomials 



GF was supported by the DAAD scholarship for one month to visit the Katholische Universität Eichstätt where this paper was started. The support of MNiSzW Iuventus Plus grant Nr 0124/IP3/2011/71 is also gratefully acknowledged.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Mathematisch–Geographische FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

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