Positive integer powers for one type of odd order circulant matrices (Q620980)

For a circulant \(n\times n\) matrix \(C_n:=\mathrm{circ}(0,a,0,\dots,0,b)\), with \(a,b\in\mathbb{C}\) and \(n\) representing an odd natural number, the authors derive a closed form of the entries of the matrix power \(C^q_n\), \(q\in\mathbb{N}\), involving Chebyshev polynomials. Proofs are based on the eigendecomposition of \(C_n\) using the discrete Fourier transform and the theory of Chebyshev polynomials. As an example, the entries of \(C^q_3\) are stated, which have been derived by a short Maple 11 program, which is also given.