In the context of linear response theory, by means of the RecurrenceRelations Method, (RRM), we find exact solutions for the relaxationfunction and the memory function of an harmonic oscillator chain inthermal equilibrium with one end fixed. We study the analyticalproperties of these solutions and show that they can exhibitreversible-non-ergodic, irreversible-non-ergodic, andirreversible-ergodic behaviors.