For a d-tuple of commuting operators S := (S 1 , · · · , S d) ∈ B[X] d , m ∈ N and p ∈ (0, ∞), we define Q (p) m (S; u) := 0≤k≤m (−1) k m k µ ∈ N d 0 |µ| = k k! µ S µ u p. As a natural extension of the concepts of (m, p)-expansive and (m, p)-contractive for tuple of commuting operators, we introduce and study the concepts of (m, ∞)-expansive tuple and (m, ∞)-contractive tuple of commuting operators acting on a Banach space. We say that S is (m, ∞)-expansive d-tuple resp. (m, ∞)-contractive d-tuple of operators if Q (p) m (S; u) ≤ 0 ∀ u ∈ X and p → ∞ resp. Q (p) m (S; u) ≥ 0 ∀ u ∈ X and p → ∞. These concepts extend the definition of (m, ∞)-isometric tuple of bounded linear operators acting on Banach spaces was introduced and studied in [13].