Asymptotically lacunary equivalent sequences defined by ideals and modulus function

Purpose: For a non-trivial ideal I ⊂ P(N), a lacunary sequence θ = (kr), and a modulus function f , the purpose of present paper is to introduce certain new notions: I(f )-asymptotically equivalent, I(wf )-asymptotically equivalent, and I(N^fθ)-asymptotically equivalent sequences of numbers.Methods: We use an analytic method to obtain our results.Results: Certain theorems on generalized equivalent sequences by the use of ideals, lacunary sequences, and a modulus function are obtained.Conclusions: We observe that if the modulus function f satisfies limt→∞f (t)/t 0, then the notions I(w) and I(Nθ ) respectively coincide with the notions I(wf ) and I(Nfθ). However, if f is bounded, the notions I(wf ) and I(N^fθ)coincides respectively with the notions I(S) and I(Sθ ).