A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Order ðα, βÞ

Fujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China Department of Mathematics, Central University of Jammu, Bagla Suchani, Samba 181143, Jammu & Kashmir, India School of Information and Physical Sciences, The University of Newcastle, Callaghan, New South Wales 2308, Australia Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

The sequence ξ = ðξ k Þ is statistically convergent of order α to ℓ (see Çolak) if there is a complex number ℓ such that Let 0 < α ≤ β ≤ 1. We define the ðα, βÞ-density of the subset E of ℕ by provided the limit exists, where jfk ≤ n : k ∈ Egj β denotes the βth power of number of elements of E not exceeding n ( [20][21][22]).
The concept of difference sequence spaces was introduced in [38] and further generalized in [39].
Let M = ðI k Þ be a Musielak-Orlicz function, u = ðu k Þ be a bounded sequence of positive real numbers, and 0 < α ≤ β ≤ 1. We define the following sequence spaces in the present paper If we take MðξÞ = ξ, the above spaces reduces to I − N

−γ
Journal of Function Spaces The following inequality will be used in the proceeding results.
for all k and r k , s k ∈ ℂ. Also jrj u k ≤ max ð1, jrj H Þ for all r ∈ ℂ.

Main Results
In this section, we study topological properties and prove some inclusion relations. In what follows, we will take M = ðI k Þ a Musielak-Orlicz function and u = ðu k Þ a bounded sequence of positive real numbers.