HIGHER ORDER DIFFERENCES ON ARBITRARY DISCRETE TIME SCALES AND RELATED GENERATING FUNCTIONS

Derek Boeckner, Scott Gensler

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we uncover some fundamental relationships involving the weights one needs to calculate the n-th difference of a function on an arbitrary discrete time scale. One of several interesting results we obtain is a formula that allows one to calculate the value of any desired finite difference coefficient directly from the graininess function for the time scale under consideration. This foundational work beautifully combines analysis, algebra, and combinatorics to obtain this and other interesting results. Some of the other interesting results include (i) for a fixed n, the n-th finite difference coefficients sum to 0 and (ii) there are nice and useful generating functions that encode various sequences of finite difference coefficients. Throughout we show the results coincide with well-known results in the special cases where the time scales are either quantum time scales or time scales with constant graininess.

Original languageEnglish (US)
Pages (from-to)431-443
Number of pages13
JournalRocky Mountain Journal of Mathematics
Volume52
Issue number2
DOIs
StatePublished - Apr 2022

Keywords

  • difference operators
  • generating functions
  • h-calculus
  • higher order finite differences
  • q-calculus
  • quantum time scale
  • Time scales

ASJC Scopus subject areas

  • Mathematics(all)

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