Detail publikačního výsledku

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

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

Originální název

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

Anglický název

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

Druh

Článek WoS

Originální abstrakt

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.

Anglický abstrakt

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.

Klíčová slova

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

Klíčová slova v angličtině

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

Autoři

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

Rok RIV

2021

Vydáno

10.09.2020

Nakladatel

Springer Nature

ISSN

1687-1847

Periodikum

Advances in Difference Equations

Svazek

2020

Číslo

1

Stát

Spojené státy americké

Strany od

1

Strany do

13

Strany počet

13

URL

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

Plný text v Digitální knihovně

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"
}