Publication detail

Note on some representations of general solutions to homogeneous linear difference equations

STEVIČ, S. IRIČANIN, B. KOSMALA, W. ŠMARDA, Z.

Original Title

Note on some representations of general solutions to homogeneous linear difference equations

Type

journal article in Web of Science

Language

English

Original Abstract

It is known that every solution to the second-order difference equation x(n) = x(n-1) + x(n-2) = 0, n >= 2, can be written in the following form x(n) = x(0)f(n-1) + x(1)f(n), where fn is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

Keywords

Homogeneous linear difference equation with constant coefficients; General solution; Representation of solutions; Fibonacci sequence

Authors

STEVIČ, S.; IRIČANIN, B.; KOSMALA, W.; ŠMARDA, Z.

Released

10. 9. 2020

Publisher

Springer Nature

ISBN

1687-1847

Periodical

Advances in Difference Equations

Year of study

2020

Number

1

State

United States of America

Pages from

1

Pages to

13

Pages count

13

URL

https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02944-y

Full text in the Digital Library

http://hdl.handle.net/11012/196552

BibTex

@article{BUT165050,
  author="Stevo {Stevič} and Bratislav {Iričanin} and Witold {Kosmala} and Zdeněk {Šmarda}",
  title="Note on some representations of general solutions to homogeneous linear difference equations",
  journal="Advances in Difference Equations",
  year="2020",
  volume="2020",
  number="1",
  pages="1--13",
  doi="10.1186/s13662-020-02944-y",
  issn="1687-1847",
  url="https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02944-y"
}