Lacunary Arithmetic Statistical Convergence
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Lacunary Arithmetic Statistical Convergence Taja Yaying1 • Bipan Hazarika2,3
Received: 14 December 2017 / Revised: 2 May 2018 / Accepted: 18 January 2020 Ó The National Academy of Sciences, India 2020
Abstract A lacunary sequence is an increasing integer sequence h ¼ ðkr Þ such that kr kr1 ! 1 as r ! 1: In this article, we introduce arithmetic statistically convergent sequence space ASC and lacunary arithmetic statistically convergent sequence space ASCh and study some inclusion properties between the two spaces. Finally, we introduce lacunary arithmetic statistical continuity and establish some interesting results. Keywords Lacunary sequence Statistical convergence Arithmetic convergence Mathematics Subject Classification 40A05 40A99 46A70 46A99
Introduction A sequence x ¼ ðxm Þ is called arithmetically convergent if for each e [ 0 there is an integer n such that for every integer m we have xm xhm;ni \e; where the symbol hm; ni denotes the greatest common divisor of two integers m and n. We denote the sequence space of all arithmetic & Bipan Hazarika [email protected] Taja Yaying [email protected] 1
Department of Mathematics, Dera Natung Govt. College, Itanagar, Arunachal Pradesh 791111, India
2
Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India
3
Department of Mathematics, Gauhati University, Guwahati, Assam 781014, India
convergent sequence by AC. The idea of arithmetic convergence was introduced by Ruckle [1]. The studies on arithmetic convergence and related results can be found in [1–5]. By a lacunary sequence, we mean an increasing integer sequence h ¼ ðkr Þ such that hr ¼ kr kr1 ! 1 as r ! 1. In this paper, the intervals determined by h will be denoted by kr Ir ¼ ðkr1 ; kr and also the ratio kr1 ; r 1; k0 [ 0 will be denoted by qr . The space of lacunary convergent sequence Nh was defined by Freedman [6] as follows: ( ) 1X Nh ¼ x ¼ ðxm Þ 2 w : lim jxm lj ¼ 0 for some l : r!1 hr m2Ir The space Nh is a BK-space with the norm 1X k xkNh ¼ sup jxm j: r hr m2I r The notion of lacunary convergence has been investigated by C ¸ olak [7], Fridy and Orhan [8, 9], Li [10], Tripathy and Et [11] and many others in the recent past. The concept of statistical convergence was introduced by Fast [12], and later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [13], Connor [14], Fridy and Orhan [9], Sˇala´t [15] and many other authors. A sequence x ¼ ðxm Þ is said to be statistically convergent to the number L if for every e [ 0; 1 lim jfm t : jxm Lj egj ¼ 0; t
t!1
where the vertical bars indicate the number of elements in the enclosed set. The main purpose of this paper is to introduce arithmetic statistical convergence and lacunary arithmetic statistical
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convergence and to study some inclusion properties between these notions. We also establish some sequential properties of lacunary arithmetic statistic
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