On a three term linear difference equation with complex coefficients

Abstract

The paper discusses some known asymptotic stability conditions
for the difference equation
$$

x_n = \alpha x_{n -1} + \beta x_{n−k},              n = 0, 1, 2, . . .

$$

with complex numbers $\alpha$, $\beta$ . As the main result, we show that the system of necessary and sufficient stability conditions, derived in a previous work, can be significantly simplified.