Circulant matrices with a special (geometric and arithmetic) sequence

In this paper, we improve the formulations of Theorem 3.8 [3] and Theorem 2.6 [1] adding some assumptions in the formulations of these theorems and we determine the Moore-Penrose inverses and the group inverses of circulant matrices with geometric and arithmetic sequence which do not satisfy these assumptions. The Moore-Penrose inverses and the group inverses of circulant matrices with constant sequence are also derived. The obtained results are illustrated by examples.