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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216305%3A26220%2F20%3APU137156" target="_blank" >RIV/00216305:26220/20:PU137156 - isvavai.cz</a>

Výsledek na webu

<a href="https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02944-y" target="_blank" >https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-020-02944-y</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1186/s13662-020-02944-y" target="_blank" >10.1186/s13662-020-02944-y</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Advances in Difference Equations

ISSN

1687-1839

e-ISSN

1687-1847

Svazek periodika

2020

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

13

Strana od-do

1-13

Kód UT WoS článku

000571752300003

EID výsledku v databázi Scopus

2-s2.0-85091356448