Explicit solutions of non-homogeneous difference equations from finite potent endomorphisms

Abstract

The aim of this work is to show the consistency of all systems of non-homogeneous linear difference equations of the form $\phi (x_{n+1})=x_n+v_0$, where $\phi \in {\mathrm{End}}_k(V)$ is a finite potent endomorphism of an arbitrary vector space V and $v_0\in V$. An algorithm to compute the set of solutions of these systems is given. In particular, the method offered is valid for computing the explicit solutions of the system of non-homogeneous difference equations $A(x_{n+1})=x_n+b$, with A being a finite square matrix.

Keywords:

• Difference equation
• finite potent endomorphism
• Drazin inverse
• finite square matrix

2010 Mathematics Subject Classifications:

• 39A06
• 47B39
• 15A09

Acknowledgments

The author would like to thank the anonymous reviewers for their valuable comments to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the Spanish Government research projects MTM2015-66760-P and PGC2018-099599-B-I00 and the Regional Government of Castile and Leon research project J416/463AC03.