On some generalized q -di ﬀ erence sequence spaces

: In this study, we construct the spaces of q -di ﬀ erence sequences of order m . We obtain some inclusion relations, topological properties, Schauder basis and alpha, beta and gamma duals of the newly deﬁned spaces. We characterize certain matrix classes from the newly deﬁned spaces to any one of the spaces c 0 , c , (cid:96) ∞ and (cid:96) p .


Introduction
A sequence space is defined as a vector subspace of ω, where ω is the set of all K-valued sequences, where K denotes R or C. Some of the well-known examples of classical sequence space are the set of all bounded sequences ∞ , null sequences c 0 , convergent sequences c and p-absolutely summable sequences p , where 1 ≤ p < ∞. A BK-space is a Banach sequence space with continuous coordinates. The space p is BK-space accompanied by the norm u p = ( k |u k | p ) 1/p . Set A = (a nk ) N 0 ×N 0 be an infinite matrix with real or complex elements. We will denote by A n = (a nk ) the sequence in the n th row of A for every n ∈ N ∪ {0}. For x = (x k ) ∈ ω, the A-transform of x is defined as the sequence Ax = ((Ax) n ) ∞ n=0 , where (Ax) n = k a nk x k provided the series on the right side converges for each n ∈ N. Furthermore, the sequence x is called A−summable to the number l if (Ax) n → l, as n → ∞. In that case, we write x → l(A) where l is called the A−limit of x. Define X, Y be two sequence spaces and A be an infinite matrix. Then, we call A a matrix mapping from X into Y, if Ax exists and is in Y for every sequence x = (x k ) ∈ X. The class of all infinite matrices that map X into Y will be denoted by (X, Y).
For a sequence space E, we call E A the matrix domain of an infinite matrix A if Then A is called a regular matrix. Several authors in the literature have constructed sequence spaces using the domain of some special matrices. For instance, one may refer to these nice papers and summability books [1][2][3][4][5][6][7].

q-sequence spaces
This paper focuses on the q-analogue of difference matrices and seeks to obtain new results related to the q-analogue.
We start with q-integer definition generated by [8]. A q-integer is defined by where q-factorial [u]! of u is given by [k], (u = 1, 2, 3, ...), From the definition of q-binomial coefficients, we have The last formula is called Gauss's q-binomial formula.
Let us now introduce the q-difference matrix definition, following [9]. We first define q-operator by where u = (u 0 , u 1 , u 2 , ...). This operator leads directly to the q-binomial coefficients via iteration, and this matrix can be explicitly represented as The primary purpose of this paper is to define a new sequence space using the ∆ m q operator and to examine this sequence space.

q-difference sequence spaces
The q-analogue of Cesàro sequence spaces were defined by Demiriz and Ş ahin [10] and Yaying et al. [11]. Then, Yaying et al. [12,13] studied over (p , q)-analogue of Euler sequence spaces and qanalogue of Catalan sequence spaces. Recently, Alotaibi et al. [14] and Yaying et al. [15,16] introduced q-difference sequence spaces of the second order. For other studies on q-analogue of sequence spaces, you can refer to the references [17,18].
Let D = ∞ , c 0 , c. The concept of a difference sequence space was introduced by Kizmaz [19], who studied the difference sequence spaces D(∆), where In the past, several authors studied matrix transformations on sequence spaces that are the matrix domain of the difference operator, or of the matrices of some classical methods of summability in different sequence spaces, for instance we refer to [20][21][22][23][24] and references therein.
Recently, Altay [24] introduced the spaces D(∆ m ) as follows: In this section, we introduce the spaces p (∆ m q ) as a generalization spaces p (∆ m ). Now let's give the sequence space p (∆ m q ) as the set of all sequences such that ∆ m q -transforms of them are in the space p , that is It is easy to check that when q = 1, the sequence space p (∆ m q ) reduces to the ordinary difference sequence space p (∆ m ) as studied by Altay [24].
In the notation of (1.1), we can redefine the space p (∆ m q ) by , which will be frequently used, as the ∆ m q -transform of a sequence Wilansky's Theorem 4.3.12 of [25, p.63] states that X is a BK-space and Λ is a triangle, then X Λ is also a BK-space endowed with the norm x X Λ = Λx X . It is easily seen that the p (∆ m q ) set becomes a linear space with the coordinatewise addition and scalar multiplication, which is a BK-space with the norm u p (∆ m q ) = ∆ m q u p .
Theorem 3.1. The p (∆ m q ) sequence space is linearly isomorphic to p .
Proof. The transformation Ψ can be defined with (3.2) notation from p (∆ m q ) to p by u → v = Ψu. Clearly, Ψ is a linear bijection and norm preserving.
and finally where |v| = (|v k |) ∞ k=0 . Now, we discuss some inclusion relations concerning with the space p (∆ m q ). Theorem 3.3. Let 0 < q < 1. The inclusion p p (∆ m q ) holds. Proof. It is fairly easy to see that the space p ⊂ p (∆ m q ). To prove the strictness part, we consider the sequence (r k ) = (k) for all k ∈ N ∪ {0}. Then, r is not a sequence in p . On the other hand from Eq (2.3) Since it is convergent, this means that ∆ m q r ∈ p and as a result r ∈ p (∆ m q ). Hence, p p (∆ m q ) holds.
A sequence Schauder basis for a linear metric space X is a sequence (u k ) ⊂ X with the property that for every u ∈ X, there exists a unique sequence (α k ) of scalars such that u − n k=1 α k u k → 0, (n → ∞).
If we take into consideration the fact that the matrix domain X A of a normed sequence space X has a basis if and only if X has a basis whenever A = (a nk ) N 0 ×N 0 is a triangle. Then, we have: is a Schauder basis for the space p (∆ m q ). It is well-known that a space which has a Schauder basis is separable, then we can give following corollary: Corollary 3.5. The sequence space p (∆ m q ) for 1 ≤ p < ∞ is separable.

α-, βand γ-duals
In this section, our next goal is to state and prove the theorems determining the α-, βand γ-duals of our new sequence spaces. The following will assume that p * is the conjugate of p, that is, p −1 +p * −1 = 1.
The α-, βand γ-duals of a sequence space U are denoted by U α , U β and U γ , respectively, and are defined by U α = {u = (u j ) ∈ w : ua = (u j a j ) ∈ 1 for all a = (a j ) ∈ U}, U β = {u = (u j ) ∈ w : ua = (u j a j ) ∈ cs for all a = (a j ) ∈ U}, First, let's give the Lemmas used in the proof of the Theorems we will give in this section: where the sequence A = (a i j ) N 0 ×N 0 is defined by ux = (u n x n ) ∈ 1 for x ∈ p (∆ m q ) if and only if Av ∈ 1 for v ∈ p . That is A ∈ ( p : 1 ). Hence, by Lemma 4.1 from (4.1), it is concluded that p (∆ m q ) α = c q .

(4.5)
Define the sets b 1 and b 2 as follows Then, Proof. We give the proof only for the case β-dual. Consider the equation for any n ∈ N. Then au = (a j u j ) ∈ cs for u ∈ p (∆ m q ) if and only if Dv ∈ c for v ∈ p . That is D ∈ ( p : c). Hence, by Lemma 4.2 from (4.2) and (4.3), it is deduced that

Matrix transformations
Let µ ∈ {c 0 , c, ∞ , p }. In this section we will characterize the spaces let p (∆ m q ) : µ and µ : p (∆ m q ) . Theorem 5.1. Define, for all k, n ∈ N, elements of infinity matrices U = (u nk ) and In this case U ∈ ( p (∆ m q ) : µ) if and only if for all n ∈ N, (u nk ) ∞ k=0 ∈ { p (∆ m q )} β and V ∈ ( p : µ). Proof. Let µ be a sequence space. Then, U = (u nk ) and V = (v nk ) satisfy the condition in (5.1). Also, the spaces p (∆ m q ) and p are linearly isomorphic, as shown in Theorem 3.1. Let U ∈ ( p (∆ m q ) : µ) and y = (y k ) ∈ p . Since (u nk ) ∞ k=0 ∈ b 1 ∩ b 2 , we have {δ nk } ∞ k=0 ∈ p for all n ∈ N ∪ {0}. Thus, ∆y exists and we have k v nk y k = k u nk x k for all n ∈ N ∪ {0}.
Corollary 5.4. Let U = (u nk ) be an infinite matrix. Then, by Theorem 5.1, the following conditions hold:

Conclusions
The theory of the q-analogue plays a significant role in various fields of mathematical, physical, and engineering sciences. Due to its vast applications in diverse fields of mathematics, several studies related to q-calculus can be found in the literature.
Recently, the construction of sequence spaces using q-calculus has been realized. The difference matrix is the most commonly used matrix in summability theory. In this study, we use the q-analogue version of the difference matrix of order m, thus providing new results.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.