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

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

—

Continuities

S - Specificky vyzkum na vysokych skolach

Publication year

2020

Confidentiality

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

Name of the periodical

Advances in Difference Equations

ISSN

1687-1839

e-ISSN

1687-1847

Volume of the periodical

2020

Issue of the periodical within the volume

1

Country of publishing house

US - UNITED STATES

Number of pages

13

Pages from-to

1-13

UT code for WoS article

000571752300003

EID of the result in the Scopus database

2-s2.0-85091356448