Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

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.

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Acknowledgements

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.

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Correspondence to Galina Filipuk.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Filipuk, G., Hilger, S. Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials. Results. Math. 64, 13–35 (2013). https://doi.org/10.1007/s00025-012-0291-5

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Mathematics Subject Classification (2000)

  • 26E70
  • 33C47

Keywords

  • Time scales
  • linear graininess time scales
  • ladder operators
  • orthogonal polynomials