Some normed binomial difference sequence spaces related to the ℓ p $\ell_{p}$ spaces

Abstract

The aim of this paper is to introduce the normed binomial sequence spaces b r,s p (∇) by combining the binomial transformation and difference operator, where 1 ≤ p ≤ ∞. We prove that these spaces are linearly isomorphic to the spaces p and ∞ , respectively. Furthermore, we compute Schauder bases and the α-, β-and γ -duals of these sequence spaces.

Binomial difference sequence spaces of fractional order

“…Let denote the class of all matrices such that . The matrix domain approach has been employed by Başarir and Kara [ 6 – 10 ], Kara and İlkhan [ 19 – 21 ], Polat and Başar [ 26 ], Song and Meng [ 23 – 25 , 27 ], and many others to introduce new sequence spaces.…”

On some binomial B ( m ) $B^{(m)}$ -difference sequence spaces

“…Since then, many authors have studied further generalization of the difference sequence spaces [4–7]. Moreover, Altay and Polat [8], Başarir and Kara [9–13], Başarir, Kara and Konca [14], Kara [15], Kara and İlkhan [16, 17], Polat and Başar [18], Song and Meng [19] and many others have studied new sequence spaces from matrix point of view that represent difference operators.…”