Eigenpairs of some imperfect pentadiagonal Toeplitz matrices

In this paper we consider pentadiagonal (n+1)×(n+1) matrices with two sub-diagonals above and below the main diagonal at distances k and ℓ from the main diagonal where 1≤k<ℓ≤n. We give explicit formulae for the eigenpairs of imperfect Toeplitz matrices in the case when (n+1)/2≤k. Imperfectness means that the first and last k elements of the main diagonal differ from the elements in the middle.