Hokkaido Mathematical Journal Vol. 35 (2006) p. 197–213 Strongly almost (V, λ)(Δ r )-summable sequences defined by Orlicz functions Mikail Et, Lee Peng Yee and Binod Chandra Tripathy (Received August 17, 2004) Abstract. The purpose of this paper is to introduce the space of sequences that are strongly almost (V, λ)(Δ r )-summable with respect to an Orlicz function. We give some relations related to these sequence spaces. We also show that the space [ ˆ V , λ, M](Δ r ) ∩ ℓ∞(Δ r ) may be represented as a ˆ s λ (Δ r ) ∩ ℓ∞(Δ r ) space. Key words : Almost (V, λ)-summability, Almost statistical convergence, Orlicz function. 1. Introduction Let w be the set of all sequences of real or complex numbers and ℓ ∞ , c and c 0 be respectively the Banach spaces of bounded, convergent and null sequences x =(x k ) with the usual norm ‖x‖ = sup |x k |, where k ∈ N = {1, 2,...}, the set of positive integers. The difference sequence spaces was introduced by Kızmaz [10] and the concept was generalized by Et and C ¸ olak [4] as follows: X (Δ r )= {x ∈ w :Δ r x ∈ X }, for X = ℓ ∞ , c and c 0 , where r ∈ N,Δ 0 x = x,Δx =(x k - x k+1 ), Δ r x = (Δ r-1 x k - Δ r-1 x k+1 ), and so Δ r x k = ∑ r v=0 (-1) v ( r v ) x k+v . These sequence spaces are BK-spaces with the norm ‖x‖ Δ = ∑ r i=1 |x i | + ‖Δ r x‖ ∞ . A sequence x ∈ ℓ ∞ is said to be almost convergent if all its Banach limits coincide and the set of all almost convergent sequences is denoted by ˆ c. Lorentz [14] proved that x ∈ ˆ c if and only if lim n (1/n) ∑ n k=1 x k+m exists uniformly in m. Several authors including Duran [2], King [9], Nanda [19], Et and Basarir [3], Malkowsky and Savas [17] and Altınok et al. [1] have stud- ied almost convergent sequences. Maddox [15], [16] has defined x to be strongly almost convergent to a number ℓ if 2000 Mathematics Subject Classification : 40A05, 40C05, 46A45.