On Some Sequence Spaces via q -Pascal Matrix and Its Geometric Properties

: We develop some new sequence spaces (cid:96) p ( P ( q )) and (cid:96) ∞ ( P ( q )) by using q -Pascal matrix P ( q ) . We discuss some topological properties of the newly deﬁned spaces, obtain the Schauder basis for the space (cid:96) p ( P ( q )) and determine the Alpha-( α -), Beta-( β -) and Gamma-( γ -) duals of the newly deﬁned spaces. We characterize a certain class ( (cid:96) p ( P ( q )) , X ) of inﬁnite matrices, where X ∈ { (cid:96) ∞ , c , c 0 } . Furthermore, utilizing the proposed results, we characterize certain other classes of inﬁnite matrices. We also examine some geometric properties, like the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and Banach–Saks-type p property of the spaces (cid:96) p ( P ( q )) and (cid:96) ∞ ( P ( q )) .


Introduction and Preliminaries
The theory of the q-analogue, as the nomenclature suggests, deals with expanding the classical mathematical concept to a more generalized one by utilizing a new parameter q.The generalized expression, i.e., q-analogue reduces to the original expression as q → 1 − .Although the history of the q-analogue dates back to Euler, the true application of the q-analogue in developing q-differentiation and q-integration was architected by Jackson [1].This concept of generalization is widely accepted by the mathematical community, and as a result, several applications of q-analogues in different branches of mathematics, like algebra, combinatorics, integro-differential equations, approximation theory, special functions, hypergeometric functions, etc., are witnessed in the literature.Indeed, the application of q-theory has also been spotted in the field of summability and sequence spaces.For instances, Demiriz and Şahin [2] and Yaying et al. [3] developed the q-analogue of the well-known Cesàro sequence spaces.Mursaleen et al. [4] discussed a regular summability method by using q-statistical convergence.They further obtained a condition for q-statistical convergent sequences to be Cesàro summable.
Apparently, the relations ( 0 0 ) q = ( v 0 ) q = ( v v ) q = 1 and ( v v−k ) q = ( v k ) q hold true for the q-binomial coefficient ( v k ) q .We recommend the monograph [5] for a basic idea of q-theory.

Sequence Space
The set ω of all real-or complex-valued sequences forms a vector space with the algebraic operations addition (+) and scalar multiplication (•) defined for the vectors y = (y n ), z = (z n ) ∈ ω and real or complex scalar α by Any vector subspace of ω is known as a sequence space.The sets p (0 < p < ∞) of all absolutely p-summable sequences, c 0 of all null sequences, c of all convergent sequences, and ∞ of all bounded sequences are some examples of basic sequence spaces.Let be the spaces of all sequences with bounded partial sums and summable sequences, respectively.
A Banach sequence space with continuous coordinates is called a BK-space.The spaces p (1 ≤ p < ∞) and ∞ are BK-spaces normed by respectively.For 0 < p < 1, the space p is a complete p-normed space due to p-norm ) be any infinite matrix with real entries.Let G n = (g n,v ) ∞ v=0 , i.e., G n denotes the n th row of the matrix G.For any y = (y v ) ∈ ω, the sequence is called G-transform of y, where we presume that the infinite sum exists for each n ∈ N 0 .Let X, Y ⊂ ω.Then, G corresponds a matrix mapping from X to Y if Gy ∈ Y for all y ∈ X.The family of all such matrices from X to Y is denoted by (X, Y).Further, the domain X G of the matrix G in a sequence space X is given by X G = {y ∈ ω : Gy ∈ X} and is a sequence space.In addition, if G is a triangle matrix and X is a BK-space, then X G is a BK-space normed by y X G = Gy X .The monographs [6,7] are worth mentioning for detailed studies concerning the domain of notable triangles in classical sequence spaces.We also mention the papers [8,9] so that readers may glean brief insight into some interesting types of summable sequence spaces.
Lately, the scholarly publications have seen an emergence of the q-analogue of wellknown sequence spaces.For instances, the domain X(C(q)) := X C(q) for X ∈ { p , c 0 , c, ∞ } are discussed by Demiriz and Şahin [2] and Yaying et al. [3], defined via the q-Cesàro matrix C(q) = (c q nv ) n,v∈N 0 given by Quite recently, some topological and geometric properties of the spaces ( p ) C(q) and ( ∞ ) C(q) have been studied by Yılmaz and Akdemir [10].Alotaibi et al. [11] engineered the sequence spaces p (∇ 2 q ) := ( p ) ∇ 2 q and ∞ (∇ 2 q ) := ( ∞ ) ∇ 2 q of the operator ∇ 2 q in p and ∞ , respectively.We recall that a sequence space X exhibits symmetry (as defined in [12]) if y π(n) belongs to X for any (y n ) in X, where π(n) represents a permutation on N 0 .Alotaibi et al. [11] proved that the space ∞ (∇ 2 q ) does not exhibit symmetricity.The theory of q-sequence spaces is further strengthened by the introduction of q-Catalan sequence space X(C(q)) = X C(q) by Yaying et al. [13] for X ∈ {c, c 0 }, and C(q) = cq nv n,v∈N 0 is defined by where c(q) = (c v (q)) v∈N 0 is a sequence of q-Catalan numbers.

Pascal Matrix and Motivation
The Pascal matrix P = ( pn,v ) n,v∈N 0 is defined by (see [14,15]) The matrix domains c 0 (P ) = (c 0 ) P and c(P ) = c P were investigated by Polat [16].These domains were further generalized by Aydın and Polat [14] by introducing Pascal difference sequence spaces c 0 (P∇) = (c 0 ) P∇ and c(P∇) = c P∇ , where ∇ is the first-order backward difference operator.Yaying and Başar [17] recently studied the sequence space X(G), where G is the product of the Lambda (Λ) matrix and Pascal matrix, i.e., G = ΛP and X is any of the spaces p , c 0 , c or ∞ .
Inspired by the above studies, we develop q-Pascal sequence spaces p (P (q)) and ∞ (P (q)) and explore several properties of these spaces.
2. The Spaces p (P (q)) and ∞ (P (q)) Define the spaces p (P (q)) and ∞ (P (q)) by Here, the sequence is termed the P (q)-transform of y = (y n ).Thus, an equivalent definition of the spaces p (P (q)) and ∞ (P (q)) may be given as It is easy to observe that when q → 1 − , the spaces p (P (q)) and ∞ (P (q)) are reduced to the spaces p (P ) = ( p ) P and ∞ (P ) = ( ∞ ) P .

Definition 3 ([19]
).The inverse of the q-Pascal matrix P (q) is given by the matrix as follows: The Q(q)-transform or {P (q)} −1 -transform of the sequence z = (z n ) is given by the sequence y = (y n ), where for each n ∈ N 0 .Throughout the article, unless stated otherwise, we keep in mind that the sequence z is the P (q)-transform of the sequence y, or equivalently, the sequence y is the {P (q)} −1 -transform of the sequence z.
Theorem 1.Each of the following statements holds true: 1.
The space p (P (q)) is a complete p-normed space for 0 < p < 1 due to the p-norm

2.
The space p (P (q)) is a BK-space for 1 ≤ p < ∞ normed by

3.
The space ∞ (P (q)) is a BK-space normed by y ∞ (P (q)) = P (q)y ∞ = sup Proof.It is known that the spaces p for 1 < p < ∞ and ∞ are BK-spaces equipped by their natural norms defined in (1) and P (q) is a triangle.Thus, Parts ( 2) and ( 3) follow immediately by using Wilansky's work [20] (Theorem 4.3.2).
A similar argument holds for the complete p-normed space p (P (q)) (0 < p ≤ 1).
Let X denote either the space p or ∞ .Consider the mapping L defined by It is noted that P (q) is the matrix representation of the operator L. Additionally, P (q) is a triangle.As a result of this fact, the mapping L is the norm or p-norm, preserving the linear bijection.Thus, X(P (q)) is linearly isomorphic to X.That is, p (P (q)) ∼ = p and It is known from Theorem 2.3 of Jarrah and Malkowsky [21] that if G is a triangle matrix, then a normed linear space X G has a basis iff X has a basis.As a result of this fact, we have the following corollary: Then, s (0) (q), s (1) (q), s (2) (q), . . . is the basis of the space p (P (q)) and each y ∈ p (P (q)) is uniquely given by Remark 1.There is no basis for the space ∞ (P (q)).
3. Some Duals of the Spaces p (P (q)) and ∞ (P (q)) Definition 5.The α-, β-and γ-duals X α , X β and X γ of any X ⊂ ω are defined by Let N denote the family of all finite subsets of N 0 , and p be the complement of p, i.e., 1/p + 1/p = 1.We recall certain lemmas that are important for determining the duals.Lemma 1 ([22-24]).Let G = (g n,v ) be an infinite matrix.Then, each of the following statements holds true: sup 4) and ( 8) hold.
Theorem 2. Define the sets λ 1 , λ 2 , and λ 3 by Then, Proof.Consider the triangle D = (d n,v ) n,v∈N 0 defined by Then, the following relation holds true: This clearly indicates that cy = (c n y n ) ∈ 1 whenever y ∈ p (P (q)) iff Dz ∈ 1 whenever z ∈ p .This means that c = (c n ) ∈ p (P (q)) α iff D ∈ ( p , 1 ).As a result of this fact and together with Lemma 1/(6), we obtain that The α-dual of ∞ (P (q)) is given in a similar manner by utilizing Lemma 1/(3) in place of Lemma 1/(6).This ends the proof.Theorem 3. Define the sets µ i (i = 1, 2, 3, 4) by , Then, Proof.Consider the triangle E = (e n,v ) n,v∈N 0 defined by Then, the following relation holds true: for each n ∈ N 0 .This indicates that cy = (c n y n ) ∈ cs whenever y = (y n ) ∈ p (P (q)) iff Ez ∈ c whenever z = (z n ) ∈ p , which means that c = (c n ) ∈ p (P (q)) β iff E ∈ ( p , c).Thus, by using Lemma 1/(5), one obtains that The β-dual of ∞ (P (q)) is obtained in a similar manner by utilizing Lemma 1/(1) in place of Lemma 1/(5) in the above proof.
By proceeding in the way similar to the above given proof of the Beta-dual, we observe that the sequence cy = (c n y n ) ∈ bs whenever y = (y n ) ∈ p (P (q)) or As a consequence of this fact and together with Lemma 1/(4) and Lemma 1/(1), the γ-dual of the spaces p (P (q)) and ∞ (P (q)) may be given as follows: Theorem 4. The given results hold true:

Matrix Transformation on the Space p (P (q))
In this section, certain classes of infinite matrices ( p (P (q)), ∞ ), ( p (P (q)), c) and ( p (P (q)), c 0 ) are characterized.As a consequence, we also give a characterization of the other classes of infinite matrices by means of a theorem.
For all n, v ∈ N 0 , define the matrix G = gn,v by Let G = (g n,v ) n,v∈N 0 be an infinite matrix over the field of complex numbers.
Theorem 5. Let 1 < p < ∞.Then, G = (g n,v ) ∈ ( p (P (q)), ∞ ) iff both of the given conditions hold true: Proof.Let 1 < p < ∞ and G ∈ ( p (P (q)), ∞ ).Then, for all y ∈ p (P (q)), P (q)y exists and is contained in the space ∞ .This implies that G n ∈ p (P (q)) β , (n ∈ N 0 ), which proves the necessity of the relation (11).Now, consider the following equality for all n ∈ N 0 , obtained by using the relation (2) We recall that the matrix mappings between BK-spaces are continuous and the fact that the spaces p (P (q)) and ∞ are BK-spaces.It follows that there exists some M > 0 such that Gy ∞ ≤ M y p (P (q)) ∀y ∈ p (P (q)).Thus, by using Hölder's inequality together with the relation (13), we obtain that Conversely, assume that the given conditions hold true and choose any y = (y v ) ∈ p (P (q)).The condition (11) means that the sequence G n ∈ p (P (q)) β , which implies that Gy exists.We recall that the space p (P (q)) is linearly isomorphic to the space p .In view of this fact, we may write that the sequence z = (z n ) ∈ p .We again apply Hölder's inequality to the relation ( 13) and obtain that This proves that Gy ∈ ∞ for all y ∈ p (P (q)).That is, G ∈ ( p (P (q)), ∞ ).
Theorem 6.Let 1 < p < ∞.Then, G ∈ ( p (P (q)), c) iff ( 11) and ( 12) hold true and Proof.Let G ∈ ( p (P (q)), c).Then, for all y ∈ p (P (q)), the sequence Gy exists and belongs to the space c.Since c ⊂ ∞ , the necessity of the conditions ( 11) and ( 12) follows immediately from Theorem 5. Let e (v) be a sequence with 1 in the v th position and 0 elsewhere.Then, for y = Q(q)e (v) , we obtain by using the relation (13) that for each fixed v ∈ N 0 .This shows the necessity of the condition (14).Conversely, assume that the conditions ( 11), ( 12) and ( 14) hold true, and choose any y = (y n ) ∈ p (P (q)).Then, G n = (g n,v ) v∈N 0 ∈ p (P (q)) β for each n ∈ N 0 , and so Gy exists.Therefore, we again obtain the relation (13).Thus, the conditions (4) and ( 7) are satisfied by the matrix G, that is to say that Gz ∈ c.Thus, by using (13), we conclude that Gy ∈ c as desired.
Proof.We leave the proof as it follows immediately from Theorem 6. Theorem 8. Let X ⊂ ω, µ ∈ { ∞ , c, c 0 } and G be any infinite matrix over the complex field C. Let R be the product matrix of infinite matrices P (q) and G, i.e., R = P (q)G = (r nv ) is given by Proof.Choose any sequence y = (y n ) ∈ X.Then, it immediately follows from the relation (15) that P (q)(Gy) = (P (q)G)y = Ry.
As a result, one has Gy ∈ Y P (q) whenever y ∈ X iff Ry ∈ Y whenever y ∈ X, which means that G ∈ (X, The following corollaries are some of the many consequences of Theorem 8. Corollary 2. Let G = (g n,v ) be an infinite matrix over the complex field C, R = (r n,v ) is defined as in (15), and the spaces c 0 (P (q)) and c(P (q)) are defined as in the work of Yaying et al. [19].Then, each of the following assertions holds true: 1.
G = (g n,v ) ∈ ( p (P (q)), ∞ (P (q))) iff each of the conditions ( 11) and ( 12) holds true with r n,v in place of g n,v for all n, v ∈ N 0 .

3.
G = (g n,v ) ∈ ( p (P (q)), c 0 (P (q))) iff each of the conditions ( 11) and ( 12) holds true, and the condition (14) also holds true with α v = 0 for all v ∈ N 0 and r n,v in place of g n,v for all n, v ∈ N 0 .
Corollary 3. Let G = (g n,v ) be an infinite matrix over the complex field C, g(n, v) = ∑ n u=0 g u,v for all n, v ∈ N 0 , and cs 0 is the space consisting of series converging to zero.Then, each of the following assertions holds true: 1.
G = (g n,v ) ∈ ( p (P (q)), bs) iff each of the conditions ( 11) and ( 12) holds true with g(n, v) in place of g n,v for all n, v ∈ N 0 .

3.
G = (g n,v ) ∈ ( p (P (q)), cs 0 ) iff each of the conditions ( 11) and ( 12) holds true, and the condition (14) also holds true with α v = 0 for all v ∈ N 0 and g(n, v) in place of g n,v for all n, v ∈ N 0 .
Corollary 4. Let G = (g n,v ) be an infinite matrix over the complex field C and the spaces λ ∞ , c λ and c λ 0 are defined as in [25].Then, each of the following assertions holds true: 1. G = (g n,v ) ∈ ( p (P (q)), λ ∞ ) iff each of the conditions ( 11) and ( 12) holds true with λn,v in place of g n,v for all n, v ∈ N 0 .

3.
G = (g n,v ) ∈ ( p (P (q)), c λ 0 ) iff each of the conditions ( 11) and ( 12) holds true, and the condition (14) also holds true with α v = 0 for all v ∈ N 0 and λn,v in place of g n,v for all n, v ∈ N 0 .
5. Some Geometric Properties of the Spaces p (P (q)) and ∞ (P (q)) In this section, we examine some geometric properties like the approximation property, Dunford-Pettis property, Hahn-Banach extension property and Banach-Saks-type p property of the spaces p (P (q)) (1 ≤ p < ∞) and ∞ (P (q)).
Let X and Y be two Banach spaces.Then, a linear operator L : X → Y is compact if whenever Q is a bounded subset of X, L(Q) is a relatively compact subset of Y [26] (Definition 3.4.1).Throughout, we use the following notation: Apparently, C(X, Y) is a closed subspace of B(X, Y), for any two Banach spaces X and Y.
We recall that an operator between any two Banach spaces is said to have a finite rank if the range of the operator is finitely dimensional.Moreover, the range of a compact linear operator between any two Banach spaces is closed iff the operator has a finite rank [26] (Proposition 3.4.6).Definition 6 ([26] (Definition 3.4.26)).A Banach space X is said to have approximation property if the set of finite rank members of B(Y, X) is dense in C(Y, X) for any Banach space Y. Theorem 9 ([26] (Theorem 3.4.27)).The space p (1 ≤ p < ∞) has the approximation property.Definition 7 ([26] (Definition 3.5.1)).Let X and Y be any two Banach spaces.Then, a linear operator L : X → Y is said to be weakly compact if whenever Q is a bounded subset of X, L(Q) is a relatively weakly compact subset of Y. Definition 8 ([26] (Definition 3.5.15)).Let X and Y be any two Banach spaces.Then, a linear operator L : X → Y is said to be completely continuous (or Dunford-Pettis operator) if L(Q) is a compact subset of Y whenever Q is a weakly compact subset of X.In this case, X is said to have a Dunford-Pettis property (in short, D-P property) if each weakly compact linear operator from X to Y is completely continuous.Lemma 2 ([26] (Exercise 3.50, p. 339)).Let L be a continuous linear operator from a normed space X to a normed space Y.Then, L(Q) is weakly compact in Y whenever Q is weakly compact in X.
Theorem 10 ([26] (Art 1.9.6, p. 75) (Hahn-Banach extension theorem)).Let V be any subspace of a normed linear space X and L 0 be a bounded linear functional on V.Then, L 0 may be extended to a bounded linear functional L on X such that L 0 = L .Theorem 11 ([27]).Let V be a linear subspace of a Banach space X and L 0 : V → ∞ be a bounded linear operator.Then, L 0 may be extended to a bounded linear operator L : X → ∞ such that L 0 = L .In this case, the space ∞ is said to possess the Hahn-Banach extension property.

Definition 9 ([28]
).A Banach space X is said to possess a weak Banach-Saks property if every weakly null sequence (y n ) ∈ X contains a subsequence (y n v ) whose Cesàro transformation is norm-convergent to zero, that is, Additionally, X is said to possess the Banach-Saks property if every bounded sequence in X contains a subsequence whose Cesàro transformation is norm-convergent in X.

Definition 10 ([29]
).A Banach space X is said to possess Banach-Saks-type p if every weakly null sequence (y n ) has a subsequence (y n v ) such that, for some C > 0, n ∑ v=0 y n v ≤ C(n + 1) 1/p , for all n ∈ N 0 .. Now we turn to the main results of this section: Theorem 12.For 1 ≤ p < ∞, the space p (P (q)) has the approximation property.
Proof.Let L : X → p (P (q)) be a compact linear operator for any Banach space X.This means that for each bounded sequence y = (y n ) ∈ X, the sequence (Ly n ) has a convergent sub-sequence (Ly n v ) in p (P (q)).That is, Ly n u − Ly n v p p (P (q)) = L(y n u − y n v ) p p (P (q)) = (P (q)L)(y n u − y n v ) p p → 0 as u, v → ∞.Thus, the operator P (q)L : X → p is well defined and compact.Now, we turn our attention to the space p that has the approximation property.It follows that there exists a sequence (T n ) of finite rank bounded linear operators from X to p such that P (q)L − T n → 0 as n → ∞.As a consequence of this fact, we realize that the sequence P (q) −1 T n of bounded linear operators from X to p (P (q)) is the required sequence of a finite rank.Additionally, Ly − P (q) −1 T n y This completes the proof.
Theorem 13.The space 1 (P (q)) has the D-P property.

Conclusions
In this article, we progressed the exploration pertaining to the advancement of "sequence spaces via specific q-matrices".This advancement was accomplished through the utilization of the q-Pascal matrix P (q).Additionally, we introduced and conducted an analysis on the sequence spaces p (P (q)) and ∞ (P (q)).Furthermore, we examined a set of outcomes related to the Schauder basis, the α-, β-, and γ-duals, along with matrix transformations related to these spaces.Within the concluding section, a comprehensive investigation into various geometric properties, namely the approximation property, Dunford-Pettis property, Hahn-Banach extension property, and the Banach-Saks-type p property, was undertaken for the spaces p (P (q)) and ∞ (P (q)).
As a prospect for future research, the exploration of the domain X P (q) of the q-Pascal matrix P (q) in Maddox's space X ∈ { (p), c 0 (p), c(p), ∞ (p)} holds promise.

n ≥ n 3 .
Proceeding in a similar way to the above technique will yield two increasing sequences (n k ) and (c k ) such thatc m ∑ k=0 y n (k)e (k) p (P (q)) < m , for all n ≥ n k+1 and ∞ ∑ k=c m +1 u m (k)e (k) p (P (q)) < m .whereu m = y n m .Thus n y n ∈ B( p (P (q))) and y p (P (q)) = ∞ ∑ n=0 ∑ n k=0 pn,k y k , it follows that y p (P (q)) ≤ 1.Therefore, we haven u m (v) p ≤ n + 1.Now, using the fact that 1 ≤ (n + 1) 1/p for all n ∈ N 0 and 1 ≤ p < ∞, we obtain n