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In this paper, the gap between the nonhomogeneous linear recurrence relation and the mixed ARMA(p; q) process in Hilbert spaces is provided. We focus ourselves on the case when the ARMA(p; q) process is of Fibonacci type. Notably, involving properties of Fibonacci sequences, we reach to establish the new formulations of some characteristics of the ARMA(p, q) process. For purpose of illustration, some examples are explored. Mathematics Subject Classification: Primary: 37H10, 40A99; Secondary: 37M10, 62M10, 65Q05

Recursiveness in Hilbert spaces and application to mixed ARMA(p, q) process