Çiğdem A. Bektaş’s research while affiliated with Ankara University Faculty of Sport Sciences and other places

Statistical convergence and summability represent a significant generalization of traditional convergence for sequences of real or complex values, allowing for a broader interpretation of convergence phenomena. This concept has been extensively examined by numerous researchers using various mathematical tools and applied to different mathematical structures over time, revealing its relevance across multiple disciplines. In the present study, a generalized definition of the concepts of statistical convergence and summability, termed (△_v^m )_u-generalized weighted statistical convergence and (△_v^m )_u-generalized weighted by [¯N_t ]-summability for real sequences, is introduced using the weighted density and generalized difference operator. Based on this definition, several fundamental properties and inclusion results, obtained by differentiating the components used in the definitions, are provided.

Modifying the definition of density functions is one method used to generalise statistical convergence. In the present study, we use sequences of modulus functions and order $\alpha \in \left( 0,1\right]$ to introduce a new density. Based on this density framework, we define strong $(f_k)$-lacunary summability of order $\alpha$ and $(f_k)$-lacunary statistical convergence of order $\alpha$ for a sequence of modulus functions $(f_k)$. This concept holds an intermediate position between the usual convergence and the statistical convergence for lacunary sequences. We also establish inclusion theorems and relations between these two concepts in the study.

In this study, we introduce a novel sequence space denoted as [Formula: see text] with a fractional order [Formula: see text]. This new space is defined by the matrix [Formula: see text], which is a composition of the Euler–Riesz matrix [Formula: see text] and the fractional ordered difference operator [Formula: see text]. We explore its topological properties along with its [Formula: see text]-, [Formula: see text]-, and [Formula: see text]-duals. Furthermore, we provide characterizations for certain matrix mappings from [Formula: see text] to the sequence spaces of Maddox.

The present study introduces the notions of statistical convergence of order α and strong p -Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.

The present study introduces the notions of statistical convergence of order $\alpha$ and strongly $q-$ summability of order $\alpha$ in partial metric spaces. We examinethe inclusion relations linked to these these concepts.

In this article, we study a new generalization of the lacunary strongly convergent sequences and introduce the concept of lacunary strong convergence according to $g^k$ for sequences of complex (or real) numbers, where $g^k=g\circ g\circ\dots\circ g$ (k times) represents a composite modulus function. After that, we determine the connections of lacunary strong convergence and lacunary statistical convergence to lacunary strong convergence according to $g^k$. Furthermore, we investigate several properties of this generalization.

The present study introduces the notions of statistical convergence of order $\alpha$ and strong $p-$ Ces\`{a}ro summability of order $\alpha$ in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of $\lambda -$% statistical convergence of order $\alpha$ in partial metric spaces while providing relations linked to these sequence spaces.

In the present paper, we introduce the notion of λm-convergence and λm -boundedness. We examine the relationship of these notions between of the ordinary convergence and ordinary boundedness, respectively. We also introduce new sequence spaces and some topological properties of these spaces. Furthermore, we investigate some inclusion relations between these sequence spaces and other well known sequence spaces.

In this paper, we introduce lacunary weak statistical convergence of order α and weak Nθα-convergence for 0<α≤1. We prove some properties of these modes of convergence and give some inclusion relations.

In this paper we introduce some new sequence spaces by using a sequence of moduli F = (fk) , give some topological properties and inclusion relations related to these sequence spaces. We also give the β-dual of [ĉ, F,p]∞(Δm).