## 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 language | English (US) |
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Pages (from-to) | 431-443 |

Number of pages | 13 |

Journal | Rocky Mountain Journal of Mathematics |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2022 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics