On the constant coefficients of a certain recurrence relation: A simple proof
Milica Anđelić1, Carlos M da Fonseca2,3
1Department of Mathematics, Kuwait University, Safat 13060, Kuwait.
Heliyon
|August 30, 2021
Summary
We present a concise proof for explicit formulas in k-periodic recurrence relations. These relations are linked to determinants of Hessenberg matrices, offering new insights into their structure.
Area of Science:
- Mathematics
- Algebraic Combinatorics
- Matrix Theory
Background:
- Recurrence relations are fundamental in various mathematical fields.
- k-periodic recurrences present unique analytical challenges.
- Hessenberg matrices offer a structured framework for studying linear recurrences.
Purpose of the Study:
- To provide a simplified proof for explicit coefficient formulas of a specific 3-term recurrence.
- To connect these recurrence relations to determinants of Hessenberg matrices.
- To enhance the understanding of k-periodic recurrence structures.
Main Methods:
- Derivation of explicit formulas for coefficients.
- Utilizing properties of k-periodic sequences.
- Interpretation through determinants of Hessenberg matrix families.
Main Results:
- A short, elegant proof for the explicit coefficient formulas.
- Demonstration of the connection between the recurrence and Hessenberg matrix determinants.
- Establishment of a clear interpretative framework.
Conclusions:
- The explicit formulas for the coefficients are validated through a concise proof.
- The study highlights a natural interpretation of these recurrences via Hessenberg matrix determinants.
- This work simplifies the analysis of k-periodic recurrence relations.
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