On determinants involving second-order recurrent sequences

TL;DR

This paper evaluates determinants of matrices formed from second-order recurrent sequences, including Lucas sequences, providing explicit formulas and characteristic polynomials for specific cases.

Contribution

It derives explicit formulas for determinants involving second-order recurrence sequences and characterizes their properties, extending known results for Lucas sequences.

Findings

Explicit determinant formulas for sequences with second-order recurrences.

Determinant evaluation for matrices built from Lucas sequences.

Characteristic polynomial determination for specific matrix cases.

Abstract

Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\ge0}$ is just the Lucas sequence $(u_n(A,B))_{n\ge0}$. In this paper, we evaluate the determinants $$\det[w_{|j-k|}]_{1\le j,k\le n}\ \ \text{and}\ \ \det[w_{|j-k+1|}]_{1\le j,k\le n}.$$ In particular, we have $$\det[u_{|j-k|}(A,B)]_{1\le j,k\le n}=(-1)^{n-1}u_{n-1}(2A,(B+1)^2).$$ When $B=-1$ and $2\mid n$, we also determine the characteristic polynomial of the matrix $[w_{j+k}]_{0\le j,k\le n-1}$.