This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐ-statistically limit and ℐ-lacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences and are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every , and ,
MSC: 40A99, 40A05.
Keywords:asymptotical equivalent; ideal convergence; lacunary sequence; statistical convergence
In 1993, Marouf  presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson  extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.
In , asymptotically lacunary statistical equivalent, which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences. Later on, the extension asymptotically lacunary statistical equivalent sequences is presented (see ).
Recently, Das, Savaş and Ghosal  introduced new notions, namely ℐ-statistical convergence and ℐ-lacunary statistical convergence by using ideal.
In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts ℐ-asymptotically statistical equivalent and ℐ-asymptotically lacunary statistical equivalent. In addition to these definitions, natural inclusion theorems shall also be presented.
First, we introduce some definitions.
2 Definitions and notations
Definition 2.1 (Marouf )
Definition 2.2 (Fridy )
The next definition is natural combination of Definitions 2.1 and 2.2.
Definition 2.3 (Patterson )
Definition 2.4 ()
The following definitions and notions will be needed.
Definition 2.5 ()
Definition 2.6 ()
Throughout ℐ will stand for a proper admissible ideal of ℕ, and by sequence we always mean sequences of real numbers.
Definition 2.8 ()
Following these results, we introduce two new notions ℐ-asymptotically lacunary statistical equivalent of multiple L and strong ℐ-asymptotically lacunary equivalent of multiple L.
The following definitions are given in .
3 New definitions
The next definitions are combination of Definitions 2.1, 2.9, 2.10 and 2.11.
For , ℐ-asymptotically statistical equivalent of multiple L coincides with asymptotically statistical equivalent of multiple L, which is defined in .
For , ℐ-asymptotically lacunary statistical equivalent of multiple L coincides with asymptotically lacunary statistical equivalent of multiple L, which is defined in .
4 Main result
In this section, we state and prove the results of this article.
Consequently, we have
Part (3): Follows from (1) and (2). □
This completes the proof. □
The author declares that they have no competing interests.
Marouf, M: Asymptotic equivalence and summability. Int. J. Math. Math. Sci.. 16(4), 755–762 (1993). Publisher Full Text
Li, J: Asymptotic equivalence of sequences and summability. Int. J. Math. Math. Sci.. 20(4), 749–758 (1997). Publisher Full Text
Patterson, RF: Analogues of some fundamental theorems of summability theory. Int. J. Math. Math. Sci.. 23(1), 1–9 (2000). Publisher Full Text