Abstract
In this paper, the concept of deferred statistical cluster points of real valued sequences is dened and studied by using deferred density of the subset of natural numbers. For p(n) and q(n) satisfying certain conditions, we give some results for the set of deferred statistical cluster points Dp;q (x). We provide some counter examples regarding Dp;q (x). Also we obtain some inclusion results for Dp;q (x). At last we consider the case q(n) = (n) and p(n) = (n 1) where the sequence = f(n)g is strictly increasing sequence of positive natural numbers with (0) = 0.
Highlights
Because of δp,q(K) does not exists for all K ⊂ N, it is convenient to use upper deferred asymptotic density of K, defining by
Steinhaus [23] independently in 1951. This subject was applied in different areas of mathematics such as in number theory by P
This subject is closely related to the subject of asymptotic density of the subset of natural numbers [2] and its root goes to A
Summary
Because of δp,q(K) does not exists for all K ⊂ N, it is convenient to use upper deferred asymptotic density of K, defining by. Keywords Natural density, statistical cluster points, statistically convergent sequence ural numbers satisfying p(n) < q(n) and lim q(n) = ∞. The concept of statistical cluster points of real valued sequences was first introduced by J.A. Fridy [11]. I) if δp,q(K) exists, δp,q(K) = δp∗,q(K), ii) δp,q(K) = 0 if and only if δp∗,q(K) > 0, iii) if K ⊂ M, δp∗,q(K) ≤ δp∗,q(M ) A real valued sequence x = (xn) is deferred statistical convergent to l, if the limit lim n→∞
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