Combinatorialization of Sury and McLaughlin identities, general linear recurrences in a unified approach

TL;DR

This paper offers combinatorial proofs for identities related to Fibonacci and Lucas numbers, interprets solutions of linear recurrences as determinants, and introduces new determinantal identities.

Contribution

It unifies combinatorial proofs of identities and provides a novel determinantal approach to linear recurrences and special number sequences.

Findings

Determinantal expression of Fibonacci and Lucas numbers

Combinatorial proof of Binets formula

New determinantal identity discovered

Abstract

In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix. Also, we derive a determinantal expression of Fibonacci and Lucas numbers. We prove Binets formula for Fibonacci and Lucas numbers in a purely combinatorial way and in course of doing so, we find a determinantal identity, which we think to be new.