The paper provides the formulae connecting the Laguerre spectrum of the output signal of a linear discretetime time-invariant system to the Laguerre spectrum of its input signal and presents a standard state-space system description. The Laguerre domain system representation is meaningful when the input signal is square-summable and the system is stable. A certain type of polynomials arising in the evaluation of the output spectrum due to the presence of time delay is defined and key properties of these are investigated. The polynomials are characterized by a three-term recurrence relation that also facilitates their numerically reliable calculation.