On the stability domain of a class of linear systems of fractional order

The uploaded paper restates the known stability criterion for Δ^q y(n+1−q)=Ay(n) by defining Sq and asserting that stability holds iff all eigenvalues of A lie in Sq with O(n^{-q}) decay, and instability if some eigenvalue lies outside cl(Sq) (Theorem 3). However, it provides no proof and cites prior work, while the model gives a coherent proof via the Z-transform, the characteristic equation det(z(1−1/z)^q I−A)=0, and a mapping of the unit circle whose image recovers exactly the boundary formula in Eq. (3). This aligns with the paper’s Theorem 1 (characteristic-zeros-in-disk) and the parametric boundary Eq. (4), so the model’s solution fills the gap. Note: the paper’s integer-order “S” set for Δy(n)=Ay(n) omits a factor 2 (|z|<−cos(arg z) instead of |z|<−2 cos(arg z)), a minor error unrelated to the fractional result (see Theorem 1 and Theorem 3 statements and Eq. (4) in the PDF).