On the number of arbitrary parameters in the general solution to a weakly delayed planar linear discrete system with constant coefficients

A planar linear discrete system with constant coefficients and two delays is considered where $k∈ℤ0∞:={0,1…,∞},x:ℤ0∞→ℝ2,$⁠, m and n are fixed integers, m > n > 0 and matrices $A={aij}i,j=12$⁠, $B={bij}i,j=12$⁠, $C={cij}i,j=12$ are constant. It is assumed that the system is weakly delayed and the eigenvalues of the matrix A are real and different. The formula for a general solution of the system is well-known and depends on 2(m+1) initial values. The paper shows that this formula can be simplified to depend only on 2 arbitrary constants. A relation between the initial values and new arbitrary constants is given.