# On Generalizedp,q-Euler Matrix and Associated Sequence Spaces

Authors: Taja Yaying; Bipan Hazarika; Mohammad Mursaleen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this study, we introduce newBK-spaces bsr,tp,q and b∞r,tp,q derived by the domain of p,q-analogue Br,tp,q of the binomial matrix in the spaces ℓs and ℓ∞, respectively. We study certain topological properties and inclusion relations of these spaces. We obtain a basis for the space bsr,tp,q and obtain Köthe-Toeplitz duals of the spaces bsr,tp,q and b∞r,tp,q. We characterize certain classes of matrix mappings from the spaces bsr,tp,q and b∞r,tp,q to space μ∈ℓ∞,c,c0,ℓ1,bs,cs,cs0. Finally, we investigate certain geometric properties of the space bsr,tp,q.


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## Body

## 1. Introduction and Preliminaries

Thep,q-calculus has been a wide and interesting area of research in recent times. Several researchers have worked in the field of p,q-calculus due to its vast applications in mathematics, physics, and engineering sciences. In the field of mathematics, it is widely used by researchers in operator theory, approximation theory, hypergeometric functions, special functions, quantum algebras, combinatorics, etc. By p,q-analogue of a known mathematical expression, we mean the generalization of that expression using two independent variables p and q rather than a single variable q as in q-calculus. If we put p=1 in the p,q-analogue of a known mathematical expression, we get q-analogue of that expression. Furthermore, when q⟶1, we receive the original expression. Chakrabarti and Jagannathan [1] introduced p,q-number to generalize several forms of q-oscillator algebras. Since then, several researchers used p,q-theory in different fields of mathematics to extend the theory of single parameter q-calculus. We strictly refer to [1–8] for studies in p,q-calculus and [9] in q-calculus.


### 1.1. Notations and Definitions onp,q-Calculus

Definition 1 (see [5]).
Let0<q<p≤1. Then, twin basic number or p,q-number is defined by
(1)ipq=pi−qip−qi>0,0i=0.
Clearly, whenp=1,ipq reduces to its q version iq.Definition 2 (see [4]).
Thep,q-analogue of binomial coefficient or p,q-binomial coefficient is defined by
(2)ijpq=ipq!i−jpq!j!i≥j,0j>i,where p,q-factorial ipq of i is given by
(3)ipq!=ipqi−1pq⋯2pq1pq.Lemma 3.
Thep,q-binomial formula is defined by
(4)x⊕yi=x+yqx+pyq2x+p2y⋯qi−1x+pi−1yi≥1,1i=0=∑j=0iijpqpi−j2qj2xjyi−j.


### 1.2. Sequence Spaces

Letw denote the set of all real-valued sequences. Any linear subspace of w is called sequence space. The following are some sequence spaces which we shall be frequently used throughout this paper:
(5)ℓ∞=x=xi∈w:supi∈ℕ0xi<∞,ℓs=x=xi∈w:∑i=0∞xis<∞,1≤s<∞,c=x=xi∈w:limi⟶∞xiexists,c0=x=xi∈w:limi⟶∞xi=0,cs=x=xi∈w:limi⟶∞∑j=0ixjexists,cs0=x=xi∈w:limi⟶∞∑j=0ixj=0,and bs denotes the space of all bounded series.Here,ℕ0 denotes the set of all natural numbers including zero. The sequence spaces ℓs and ℓ∞ are Banach spaces equipped with the norms
(6)xℓs=∑i=0∞xis1/sandxℓ∞=supi∈ℕ0xi,respectively.Letλ and μ be two sequence spaces and Φ=ϕij be an infinite matrix of real entries. By Φi, we denote the ith row of the matrix Φ. We say that Φ defines a matrix mapping from λ to μ if Φx∈μ for every x=xj∈λ, where Φx=Φxi=∑j=0∞ϕijxj is Φ-transform of the sequence x. The notation λ:μ will denote the family of all matrices that map from λ to μ.The matrix domainλΦ of the matrix Φ in the space λ is defined by
(7)λΦ=x∈w:Φx∈λ,which itself is a sequence space. Using this notation, several authors in the past have constructed sequence spaces using some special matrices. For relevant literature, we refer to the papers [10–15] and textbooks [16–18]. For some recent publications dealing with the domain of triangles in classical spaces, we refer [19–28].


### 1.3. Literature Review

We give a short survey of literature concerning Euler sequence spaces. Altay and Başar [10] introduced Euler sequence space e0r=c0Er and e∞r=ℓ∞Er, obtained their α-, β-, γ-, and continuous duals, and characterized certain class of matrix mappings on the space cEr, where Er=eijr denotes the Euler matrix of order r and is defined by
(8)eijr=ij1−ri−jrj0≤j≤i,0j>i,for all i,j∈ℕ0 and 0<r<1. The Euler matrix Er is regular for 0<r<1 and is invertible with Er−1=E1/r.Altay et al. [11] introduced the Euler space esr=ℓsEr,1≤s≤∞, and obtained certain inclusion relations, Schauder basis and Köthe-Toeplitz duals of the space esr. As a natural continuation of [11], Mursaleen et al. [14] characterized various classes of matrix mappings from the space esr to other spaces and examined certain geometric properties of the space esr. Further, Altay and Polat [29] introduced Euler difference spaces e0rΔB=c0ErΔB and ecrΔB=cErΔB, where ΔB is backward difference operator defined by ΔBvj=vj−vj−1 for all j∈ℕ0. Extending these spaces, Polat and Başar [30] studied Euler difference spaces e0rΔBm,ecrΔBm, and e∞rΔBm of mthm∈ℕ order defined as the set of all sequences whose mth order backward differences are in the spaces e0r,ecr, and e∞r, respectively. Kadak and Baliarsingh [31] further generalized these spaces by introducing Euler difference spaces esrΔBq,e0rΔBq,ecrΔBq, and e∞rΔBq of fractional order q, where ΔBq is the backward fractional difference operator defined by ΔBqxj=∑i=0∞−1iΓq+1/i!Γq−i+1xj−i. Kara et al. [32] introduced paranormed Euler space erp=ℓpEr and studied its topological and geometric properties. Aftermore, Karakaya and Polat [33] studied paranormed Euler difference sequence spaces e0rΔB,p,ecrΔB,p, and e∞rΔB,p. Extending these spaces, Karakaya et al. [34] studied paranormed Euler backward difference spaces e0rΔBm,p,ecrΔBm,p, and e∞rΔBm,p of mth order. Besides, Demiriz and Çakan [35] introduced paranormed Euler difference spaces e0ru,p and ecru,p. Furthermore, Kirisci [36] introduced Euler almost null f0Er and Euler almost convergent fEr sequence spaces. Later on, Kara and Başar [37] introduced generalized difference Euler spaces e0rBm=c0Bm,ecrBm=cBm and e∞rBm=ℓ∞Bm, where Bm=bijm is a generalized difference matrix defined by
(9)bijm=mi−jrm−i+jsi−jmax0,i−m≤j≤i,00≤j≤max0,i−morj>i,and characterized certain classes of compact operators on the spaces e0rBm and e∞rBm. Meng and Mei [38] gave a further generalization of [37] by introducing Euler difference spaces e0rBvm and ecrBvm, where the difference operator Bvm=bijm,v is defined by
(10)bijm,v=mi−jrm−i+jsi−jvjmax0,i−m≤j≤i,00≤j≤max0,i−morj>i,where v=vj is a fixed sequence of nonzero real numbers. Recently, Bisgin [39, 40] introduced more generalized Euler space by defining binomial spaces bsr,t=ℓsBr,t,b0r,t=c0Br,t,bcr,t=cBr,t, and b∞r,t=ℓ∞Br,t, and Br,t=bijr,t is the binomial matrix defined by
(11)bijr,t=1r+tiijrisi−j0≤j≤i,0j>i.Meng and Song [41] further generalized these spaces by introducing binomial Bm-difference sequence spaces b0r,tBm=c0Br,tBm,bcr,tBm=cBr,tBm, and e∞rBm=ℓ∞Br,tBm. Meng and Mei [42] studied binomial backward difference sequence spaces b0r,tΔBq=c0Br,tΔBq,bcr,tΔBq=cBr,tΔBq, and b∞r,tΔBq=ℓ∞Br,tΔBq of fractional order q. Besides, Yaying and Hazarika [27] also studied binomial backward difference spaces bsr,tΔBq=ℓsBr,tΔBq of fractional order q.For0<q<1, the q-Cesàro matrix Cq=cijq [43, 44] is defined by
(12)cijq=qji+1q0≤j≤i,0j>i.Demiriz and Sahin [45] studied the domain of q-Cesàro mean in the spaces c and c0. Very recently, Yaying et al. [28] studied Banach sequence spaces Xsq and X∞q defined as the domain of q-Cesàro mean in the spaces ℓs and ℓ∞, respectively, and studied associated operator ideals.Motivated by the above studies, we generalize Euler meanEr and Binomial mean Br,t in the sense of p,q-theory to Br,tp,q and study its domain bsr,tp,q and b∞r,tp,q in the spaces ℓs and ℓ∞, respectively. We investigate some topological properties and inclusion relations of the spaces bsr,tp,q and b∞r,tp,q and obtain a basis for the space bsr,tp,q. In Section 3, we obtain the Köthe-Toeplitz duals (α-, β-, and γ-duals) of the spaces bsr,tp,q and b∞r,tp,q. In Section 4, we characterize some matrix mappings from bsr,tp,q and b∞r,tp,q spaces to space μ∈ℓ∞,c,c0,ℓ1,cs,cs0,bs. Section 5 is devoted to investigation of certain geometric properties like Banach-Saks of type s and modulus of convexity of the space bsr,tp,q.In the rest of the paper,1≤s<∞, unless stated otherwise.


## 1.1. Notations and Definitions onp,q-Calculus

Definition 1 (see [5]).
Let0<q<p≤1. Then, twin basic number or p,q-number is defined by
(1)ipq=pi−qip−qi>0,0i=0.
Clearly, whenp=1,ipq reduces to its q version iq.Definition 2 (see [4]).
Thep,q-analogue of binomial coefficient or p,q-binomial coefficient is defined by
(2)ijpq=ipq!i−jpq!j!i≥j,0j>i,where p,q-factorial ipq of i is given by
(3)ipq!=ipqi−1pq⋯2pq1pq.Lemma 3.
Thep,q-binomial formula is defined by
(4)x⊕yi=x+yqx+pyq2x+p2y⋯qi−1x+pi−1yi≥1,1i=0=∑j=0iijpqpi−j2qj2xjyi−j.


## 1.2. Sequence Spaces

Letw denote the set of all real-valued sequences. Any linear subspace of w is called sequence space. The following are some sequence spaces which we shall be frequently used throughout this paper:
(5)ℓ∞=x=xi∈w:supi∈ℕ0xi<∞,ℓs=x=xi∈w:∑i=0∞xis<∞,1≤s<∞,c=x=xi∈w:limi⟶∞xiexists,c0=x=xi∈w:limi⟶∞xi=0,cs=x=xi∈w:limi⟶∞∑j=0ixjexists,cs0=x=xi∈w:limi⟶∞∑j=0ixj=0,and bs denotes the space of all bounded series.Here,ℕ0 denotes the set of all natural numbers including zero. The sequence spaces ℓs and ℓ∞ are Banach spaces equipped with the norms
(6)xℓs=∑i=0∞xis1/sandxℓ∞=supi∈ℕ0xi,respectively.Letλ and μ be two sequence spaces and Φ=ϕij be an infinite matrix of real entries. By Φi, we denote the ith row of the matrix Φ. We say that Φ defines a matrix mapping from λ to μ if Φx∈μ for every x=xj∈λ, where Φx=Φxi=∑j=0∞ϕijxj is Φ-transform of the sequence x. The notation λ:μ will denote the family of all matrices that map from λ to μ.The matrix domainλΦ of the matrix Φ in the space λ is defined by
(7)λΦ=x∈w:Φx∈λ,which itself is a sequence space. Using this notation, several authors in the past have constructed sequence spaces using some special matrices. For relevant literature, we refer to the papers [10–15] and textbooks [16–18]. For some recent publications dealing with the domain of triangles in classical spaces, we refer [19–28].


## 1.3. Literature Review

We give a short survey of literature concerning Euler sequence spaces. Altay and Başar [10] introduced Euler sequence space e0r=c0Er and e∞r=ℓ∞Er, obtained their α-, β-, γ-, and continuous duals, and characterized certain class of matrix mappings on the space cEr, where Er=eijr denotes the Euler matrix of order r and is defined by
(8)eijr=ij1−ri−jrj0≤j≤i,0j>i,for all i,j∈ℕ0 and 0<r<1. The Euler matrix Er is regular for 0<r<1 and is invertible with Er−1=E1/r.Altay et al. [11] introduced the Euler space esr=ℓsEr,1≤s≤∞, and obtained certain inclusion relations, Schauder basis and Köthe-Toeplitz duals of the space esr. As a natural continuation of [11], Mursaleen et al. [14] characterized various classes of matrix mappings from the space esr to other spaces and examined certain geometric properties of the space esr. Further, Altay and Polat [29] introduced Euler difference spaces e0rΔB=c0ErΔB and ecrΔB=cErΔB, where ΔB is backward difference operator defined by ΔBvj=vj−vj−1 for all j∈ℕ0. Extending these spaces, Polat and Başar [30] studied Euler difference spaces e0rΔBm,ecrΔBm, and e∞rΔBm of mthm∈ℕ order defined as the set of all sequences whose mth order backward differences are in the spaces e0r,ecr, and e∞r, respectively. Kadak and Baliarsingh [31] further generalized these spaces by introducing Euler difference spaces esrΔBq,e0rΔBq,ecrΔBq, and e∞rΔBq of fractional order q, where ΔBq is the backward fractional difference operator defined by ΔBqxj=∑i=0∞−1iΓq+1/i!Γq−i+1xj−i. Kara et al. [32] introduced paranormed Euler space erp=ℓpEr and studied its topological and geometric properties. Aftermore, Karakaya and Polat [33] studied paranormed Euler difference sequence spaces e0rΔB,p,ecrΔB,p, and e∞rΔB,p. Extending these spaces, Karakaya et al. [34] studied paranormed Euler backward difference spaces e0rΔBm,p,ecrΔBm,p, and e∞rΔBm,p of mth order. Besides, Demiriz and Çakan [35] introduced paranormed Euler difference spaces e0ru,p and ecru,p. Furthermore, Kirisci [36] introduced Euler almost null f0Er and Euler almost convergent fEr sequence spaces. Later on, Kara and Başar [37] introduced generalized difference Euler spaces e0rBm=c0Bm,ecrBm=cBm and e∞rBm=ℓ∞Bm, where Bm=bijm is a generalized difference matrix defined by
(9)bijm=mi−jrm−i+jsi−jmax0,i−m≤j≤i,00≤j≤max0,i−morj>i,and characterized certain classes of compact operators on the spaces e0rBm and e∞rBm. Meng and Mei [38] gave a further generalization of [37] by introducing Euler difference spaces e0rBvm and ecrBvm, where the difference operator Bvm=bijm,v is defined by
(10)bijm,v=mi−jrm−i+jsi−jvjmax0,i−m≤j≤i,00≤j≤max0,i−morj>i,where v=vj is a fixed sequence of nonzero real numbers. Recently, Bisgin [39, 40] introduced more generalized Euler space by defining binomial spaces bsr,t=ℓsBr,t,b0r,t=c0Br,t,bcr,t=cBr,t, and b∞r,t=ℓ∞Br,t, and Br,t=bijr,t is the binomial matrix defined by
(11)bijr,t=1r+tiijrisi−j0≤j≤i,0j>i.Meng and Song [41] further generalized these spaces by introducing binomial Bm-difference sequence spaces b0r,tBm=c0Br,tBm,bcr,tBm=cBr,tBm, and e∞rBm=ℓ∞Br,tBm. Meng and Mei [42] studied binomial backward difference sequence spaces b0r,tΔBq=c0Br,tΔBq,bcr,tΔBq=cBr,tΔBq, and b∞r,tΔBq=ℓ∞Br,tΔBq of fractional order q. Besides, Yaying and Hazarika [27] also studied binomial backward difference spaces bsr,tΔBq=ℓsBr,tΔBq of fractional order q.For0<q<1, the q-Cesàro matrix Cq=cijq [43, 44] is defined by
(12)cijq=qji+1q0≤j≤i,0j>i.Demiriz and Sahin [45] studied the domain of q-Cesàro mean in the spaces c and c0. Very recently, Yaying et al. [28] studied Banach sequence spaces Xsq and X∞q defined as the domain of q-Cesàro mean in the spaces ℓs and ℓ∞, respectively, and studied associated operator ideals.Motivated by the above studies, we generalize Euler meanEr and Binomial mean Br,t in the sense of p,q-theory to Br,tp,q and study its domain bsr,tp,q and b∞r,tp,q in the spaces ℓs and ℓ∞, respectively. We investigate some topological properties and inclusion relations of the spaces bsr,tp,q and b∞r,tp,q and obtain a basis for the space bsr,tp,q. In Section 3, we obtain the Köthe-Toeplitz duals (α-, β-, and γ-duals) of the spaces bsr,tp,q and b∞r,tp,q. In Section 4, we characterize some matrix mappings from bsr,tp,q and b∞r,tp,q spaces to space μ∈ℓ∞,c,c0,ℓ1,cs,cs0,bs. Section 5 is devoted to investigation of certain geometric properties like Banach-Saks of type s and modulus of convexity of the space bsr,tp,q.In the rest of the paper,1≤s<∞, unless stated otherwise.


## 2. Generalized Euler Sequence Spacesbsr,tp,q and b∞r,tp,q

In this section, we introduce sequence spacesbsr,tp,q and b∞r,tp,q, study their topological properties and some inclusion relations, and obtain a basis for the space bsr,tp,q.Letr,t be nonnegative real numbers and 0<q<p≤1 holds, then the generalized p,q-Euler matrix Br,tp,q=bijr,t of order r,t is defined by
(13)bijr,t=1r⊕tiijpqpi−j2qj2rjti−j0≤j≤i,0j>i.One can clearly observe that the matrixBr,tp,q reduces to the binomial matrix Br,t when p=q=1. Thus, Br,tp,q generalizes binomial matrix Br,t. We may call the matrix Br,tp,q as the p,q-analogue of the binomial matrix Br,t. We also realise that when p=1, the matrix Br,tp,q reduces to its q-version Br,tq with entries ijqqj2rjti−j/r⊕tqi if 0≤j≤i and 0, otherwise. We call Br,tq as the q-analogue of the binomial matrix Br,t. Moreover, when t=1−r, then the matrix Br,tp,q reduces to Erp,q with entries 1/r⊕1−riijpqpi−j2qj2rj1−ri−j if 0≤j≤i and 0, otherwise. The generalized p,q-Euler sequence spaces bsr,tp,q and b∞r,tp,q are defined by
(14)bsr,tp,q=x=xj∈w:∑i=0∞1r⊕ti∑j=0iijpqpi−j2qj2rjti−js<∞,b∞r,tp,q=x=xj∈w:supi∈ℕ01r⊕ti∑j=0iijpqpi−j2qj2rjti−j<∞.The above sequence spaces can be redefined in the notation of (7) by
(15)bsr,tp,q=ℓsBr,tp,qandb∞r,t=ℓ∞Br,tp,q.The spacesbsr,tp,q and b∞r,tp,q reduce to the following classes of spaces in the special cases of p,q and r,t:
(1)
Whenp=1, the spaces bsr,tp,q and b∞r,tp,q reduce to q-binomial sequence spaces bsr,tq=ℓsBr,tq and b∞r,tq=ℓ∞Br,tq, respectively, which further reduce to binomial sequence spaces bsr,t and b∞r,t, respectively, when q⟶1, as studied by Bisgin [40](2)
Whenp=1 and r+t=1, the spaces bsr,tp,q and b∞r,tp,q reduce to q-Euler space esrq=ℓsErq and e∞rq=ℓ∞Erq, respectively, which further reduce to well known Euler sequence spaces esr and e∞r, respectively, when q⟶1, as studied by Altay et al. [11](3)
Whenr+t=1, the spaces bsr,tp,q and b∞r,tp,q reduce to p,q-Euler sequence spaces esrp,q=ℓsErp,q and e∞rp,q=ℓ∞Erp,qLet us define a sequencey=yi in terms of sequence x=xj by
(16)yi=Br,tp,qxi=1r⊕ti∑j=0iijpqpi−j2qj2rjti−jxj,for each i∈ℕ0. The sequence y is called Br,tp,q-transform of the sequence x. Further, on using (16), we write
(17)xi=∑j=0i−1i−jijpqqi−j2ti−jr⊕tjriqi2yj,for each i∈ℕ0.It is known that ifλ is a BK-space and Φ is a triangle then the domain λΦ of the matrix Φ in the space λ is also a BK-space equipped with the norm xλΦ=Φxλ. In the light of this, we have the following result.Theorem 4.
The sequence spacesbsr,tp,q and b∞r,tp,q are BK-spaces equipped with the norms defined by
(18)xbsr,tp,q=Br,tp,qxℓs=∑i=0∞1r⊕tpqi∑j=0iijpqpi−j2qj2rjti−jxjs1/s,xb∞r,tp,q=Br,tp,qxℓ∞=supi∈ℕ01r⊕tpqi∑j=0iijpqpi−j2qj2rjti−jxj,respectively.Proof.
The proof is a routine exercise and hence omitted.☐Theorem 5.
The sequence spacesbsr,tp,q and b∞r,tp,q are linearly isomorphic to ℓs and ℓ∞, respectively.Proof.
We provide the proof for the spacebsr,tp,q. Define the mapping T:bsr,tp,q⟶ℓs by Tx=Br,tp,qx for all x∈bsr,tp,q. It is easy to observe that T is linear and one to one. Let y=yi∈ℓs and x=xi is as defined in (17). Then, we have
(19)xbsr,tp,q=∑i=0∞1r⊕tpqi∑j=0iijpqpi−j2qj2rjti−jxjs1/s=∑i=0∞1r⊕tpqi∑j=0iijpqpi−j2qj2rjti−jxj∑k=0j−1j−kijpqqj−k2tj−kr⊕tkrjqj2yks1/s=∑i=0∞yis1/s=yℓs<∞.
Thus,x∈bsr,tp,q and the mapping T:bsr,tp,q⟶ℓs is onto and norm preserving. Hence, the space bsr,tp,q is linearly isomorphic to ℓs. This completes the proof.☐Theorem 6.
The spacebsr,tp,q,1≤s≤∞, is not a Hilbert space, except for the case s=2.Proof.
Define the sequencesx=xi and y=yi by
(20)xi=1i=0,−1itri−r⊕tqi−12ti−1i1pqriqi2i>0,yi=1i=0,−1itri+r⊕tqi−12ti−1i1pqriqi2i>0.
We realise thatBr,tp,qxi=1,1,0,0,⋯ and Br,tp,qyi=1,−1,0,0,⋯. Then
(21)x+ybsr,tp,q2+x−ybsr,tp,q2=8≠22+2/s=2xbsr,tp,q2+ybsr,tp,q2.
Thus,bsr,tp,q norm violates the parallelogram identity. Hence, bsr,tp,q is not a Hilbert space, except for the case s=2.☐Now we give certain inclusion relations related to the spacesbsr,tp,q and b∞r,tp,q.Theorem 7.
The inclusionℓs⊂bsr,tp,q,1≤s≤∞, strictly holds.Proof.
We provide proof of the inclusionℓs⊂bsr,tp,q,1≤s<∞. Let x=xi∈ℓs for 1<s<∞. Applying Hölder’s inequality, we have
(22)∑i=0∞Br,tp,qxis≤∑i=0∞∑j=0iijpqpi−j2qj2rjti−jr⊕tixjs≤∑i=0∞∑j=0iijpqpi−j2qj2rjti−jr⊕tixjs∑j=0iijpqpi−j2qj2rjti−jr⊕tis−1=∑i=0∞∑j=0iijpqpi−j2qj2rjti−jr⊕tixjs=∑j=0∞xjs∑i=j∞ijpqpi−j2qj2rjti−jr⊕ti.
Thus,xbsr,tp,qs≤Kxℓss<∞, where K=supj∈ℕ0∑i=j∞ijpqpi−j2qj2rjti−j/r⊕ti, provided K exists. This yields the fact that x∈bsr,tp,q. Thus, ℓs⊂bsr,tp,q. Similarly, we can show that ℓ1⊂b1r,tp,q.
Now consider the sequencexi=−1i, then it is easy to see that xi∈bsr,tp,q\ℓs. Hence, the inclusion ℓs⊂bsr,tp,q is strict. This completes the proof.☐Theorem 8.
The inclusionbsr,tp,q⊂bkr,tp,q,1≤s<k<∞, strictly holds.Proof.
It is known that inclusionℓs⊂ℓk holds for 1≤s<k<∞ and the mapping Br,tp,q:bsr,tp,q⟶ℓs is isomorphic, therefore, the inclusion bsr,tp,q⊂bkr,tp,q holds. To prove the strictness part, we recall that the inclusion ℓs⊂ℓk strictly holds for 1≤s<k<∞. We choose y∈ℓk\ℓs and x as defined in (17). Then, Br,tp,qx=y∈ℓk\ℓs. This implies that x∈bkr,tp,q\bsr,tp,q. Hence, the inclusion bsr,tp,q⊂bkr,tp,q is strict.☐Theorem 9.
The inclusionbsr,tp,q⊂b∞r,tp,q strictly holds.Proof.
The proof is similar to the proof of Theorem8. To show the strictness part, we consider the sequence xk=1,1,1,⋯. Then, it is clear that x∈b∞r,tp,q\bsr,tp,q. Hence, the inclusion bsr,tp,q⊂b∞r,tp,q strictly holds.☐We recall that domainλΦ of a triangle Φ in space λ has a basis if and only if λ has a basis. This statement together with Theorem 5 gives us the following result.Theorem 10.
Letξj=Br,tp,qxj for each j∈ℕ0. Define the sequence bjp,q=bijp,q of elements of the space bsr,tp,q for every fixed j∈ℕ0by(23)bijp,q=−1i−jijpqqi−j2ti−jr⊕tjriqi2j≤i,0j>i.
Then, the sequencebjp,q forms a basis for the space bsr,tp,q and every x∈bsr,tp,q can be uniquely expressed in the form x=∑j=0∞ξjbjp,q for each j∈ℕ0.


## 3. Köthe-Toeplitz Duals

In this section, we obtain Köthe-Toeplitz duals (α-, β-, and γ-duals) of the spaces bsr,tp,q and b∞r,tp,q. We omit the proofs for cases s=1 and s=∞ as these can be obtained by analogy and provide proofs for only the case 1<s<∞ in the current section. First, we recall the definitions of Köthe-Toeplitz duals.Definition 11.
The Köthe-Toeplitz duals orα−,β−, and γ-duals of subset λ⊂w are defined by
(24)λα=a=ai∈w:ax=aixi∈ℓ1forallx∈λ,λβ=a=ai∈w:ax=aixi∈csforallx∈λand,λγ=a=ai∈w:ax=aixi∈bsforallx∈λ,respectively.Quite recently, Talebi [25] obtained Köthe-Toeplitz duals of the domain of an arbitrary invertible summability matrix in ℓs space. We follow his approach to find the Köthe-Toeplitz duals of the spaces bsr,tp,q and b∞r,tp,q. In the rest of the paper, N will denote the family of all finite subsets of ℕ0 and k=s/1−s is the complement of s.Theorem 12.
Define the setsdkp,q and d∞p,q by
(25)dkp,q=a=ai∈w:supN∈N∑j=0∞∑i∈N−1i−jijpqqi−j2ti−jr⊕tjriqi2aik<∞and,d∞p,q=a=ai∈w:supj∈ℕ0∑i=0∞−1i−jijpqqi−j2ti−jr⊕tjriqi2ai<∞.
Then,b1r,tp,qα=d∞p,q,bsr,tp,qα=dkp,q and b∞r,tp,qα=d1p,q.Proof.
Let1<s<∞. Let ai∈w and y=yi be the Br,tp,q-transform of sequence x=xi. Then, from the equality (17), we have
(26)aixi=∑j=0i−1i−jijpqqi−j2ti−jr⊕tjriqi2aiyj=Gr,tp,qyi,for all i∈ℕ0, where the matrix Gr,tp,q=gijr,tp,q is defined by
(27)gijr,tp,q=−1i−jijpqqi−j2ti−jr⊕tjriqi2ai0≤j≤i,0j>i.
Applying Theorem 2.1 of [25], we immediately obtained that
(28)bsr,tp,qα=a=ai∈w:supN∈N∑j=0∞∑i∈N−1i−jijpqqi−j2ti−jr⊕tjriqi2aik<∞.
This completes the proof.☐Theorem 13.
Define the setsd1p,q,d2p,q,d3p,q, and dkp,q by
(29)d1p,q=a=al∈w:∑l=j∞−1l−jljpqql−j2tl−jr⊕tjrlql2alexistsforeachj∈ℕ0,d2p,q=a=al∈w:supi,j∈ℕ0∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2al<∞,d3p,q=a=al∈w:limi⟶∞∑j=0i∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2al=∑j=0ilimi⟶∞∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2al,dkp,q=a=al∈w:supi∈ℕ0∑j=0i∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2alk<∞.
Then,b1r,tp,qβ=d1p,q∩d2p,q,bsr,tβ=d1p,q∩dkp,q and b∞r,tp,qβ=d1p,q∩d3p,q.Proof.
Letai∈w and y=yi be the Br,tp,q-transform of sequence x=xi. Then, from the equality (17), we get
(30)∑j=0iajxj=∑j=0i∑l=0j−1j−ljlpqqj−l2tj−lr⊕tlrjqj2ylaj=∑j=0i∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2alyj,=Hr,tp,qyi,for each i∈ℕ0, where the matrix Hr,tp,q=hijr,tp,q is defined by
(31)hijr,tp,q=∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2al0≤j≤i,0j>i,for all i,j∈ℕ0.
Thus, by applying Theorem 2.2 of [25], we straightly get
(32)bsr,tp,qβ=d1p,q∩dkp,q.
This completes the proof.☐Theorem 14.
Define the setd4p,q by
(33)d4p,q=a=ai∈w:supi∈ℕ0∑j=0i∑l=ji−1l−jljpqql−j2tl−jr⊕tjrlql2al<∞.
Then,b1r,tp,qγ=d2p,q,bsr,tp,qγ=dkp,q and b∞r,tp,qγ=d4p,q.Proof.
The proof is similar to the previous theorem except that Theorem 2.3 of [25] is employed instead of Theorem 2.2 of [25].☐


## 4. Matrix Mappings

In this section, we characterize a certain class of matrix mappings from the spacesbsr,tp,q and b∞r,tp,q to space μ∈ℓ∞,c,c0,ℓ1,bs,cs,cs0. The following theorem is fundamental in our investigation.Theorem 15.
Let1≤s≤∞ and λ be an arbitrary subset of w. Then, Φ=ϕij∈bsr,tp,q:λ if and only if Ψi=ψmji∈ℓs:c for each i∈ℕ0, and Ψ=ψij∈ℓs:λ, where
(34)ψmji=0j>m,∑l=jm−1l−jljpqql−j2tl−jr⊕tjrlql2ϕil0≤j≤m,ψij=∑l=j∞−1l−jljpqql−j2tl−jr⊕tjrlql2ϕil.for all i,j∈ℕ0.Proof.
The proof is similar to the proof of Theorem 4.1 of [13]. Hence, we omit details.☐Now, using the results presented in Stielglitz and Tietz [46] together with Theorem 15, we obtain the following results:Corollary 16.
The following statements hold:(1)
Φ∈b1r,tp,q:ℓ∞ if and only if
(35)limm⟶∞ψmjiexistsforalli,j∈ℕ0,(36)supi,j∈ℕ0ψmji<∞,(37)supi,j∈ℕ0ψij<∞,(2)
Φ∈b1r,tp,q:c if and only if (35) and (36) hold, and (37) and
(38)limi⟶∞ψijexistsforallj∈ℕ0,also hold(3)
Φ∈b1r,tp,q:c0 if and only if (35) and (36) hold, and (37) and
(39)limi⟶∞ψij=0forallj∈ℕ0,also hold(4)
Φ∈b1r,tp,q:ℓ1 if and only if (35) and (36) hold, and
(40)supj∈ℕ0∑i=0∞ψij<∞,also holds(5)
Φ∈b1r,tp,q:bs if and only if (35) and (36) hold, and (37) also holds with Ψi,j instead of ψij, where Ψi,j=∑l=0iψlj(6)
Φ∈b1r,tp,q:cs if and only if (35) and (36) hold, and (37) and (38) also hold with Ψi,j instead of ψij, where Ψi,j=∑l=0iψlj(7)
Φ∈b1r,tp,q:cs0 if and only if (35) and (36) hold, and (37) and (39) also hold with Ψi,j instead of ψij, where Ψi,j=∑l=0iψljCorollary 17.
The following statements hold:(1)
Φ∈bsr,tp,q:ℓ∞ if and only if (35) holds, and(41)supm∈ℕ0∑j=0mψmjik<∞,(42)supi∈ℕ0∑j=0iψijk<∞,also hold
(2)
Φ∈bsr,tp,q:c if and only if (35) and (41) hold, and (38) and (42) also hold(3)
Φ∈bsr,tp,q:c0 if and only if (35) and (41) hold, (39) and (42) also hold(4)
Φ∈bsr,tp,q:ℓ1 if and only if (35) and (41) hold, and(43)supN∈N∑j=0∞∑i∈Nψijk<∞,also holds
(5)
Φ∈bsr,tp,q:bs if and only if (35) and (41) hold, and (42) also holds with Ψi,j instead of ψij, where Ψi,j=∑l=0iψlj(6)
Φ∈bsr,tp,q:cs if and only if (35) and (41) hold, and (38) and (42) also hold(7)
Φ∈bsr,tp,q:cs0 if and only if (35) and (41) hold, and (39) and (42) also hold with Ψi,j instead of ψij, where Ψi,j=∑l=0iψljCorollary 18.
The following statements hold:(1)
Φ∈b∞r,tp,q:ℓ∞ if and only if (35) and
(44)limm⟶∞∑j=0mψmji=∑j=0mlimm⟶∞ψmjiforeachi∈ℕ0,hold, and (42) also holds with k=1(2)
Φ∈b∞r,tp,q:c if and only if (35) and (44) hold, and (38) and
(45)limi⟶∞∑j=0iψij=∑j=0ilimi⟶∞ψij,also hold(3)
Φ∈b∞r,tp,q:c0 if and only if (35) and (44) hold, and
(46)limi⟶∞∑j=0iψij=0,also holds(4)
Φ∈b∞r,tp,q:ℓ1 if and only if (35) and (44) hold, and (43) also holds with k=1(5)
Φ∈b∞r,tp,q:bs if and only if (35) and (44) hold, and (42) also hold with k=1, and Ψi,j instead of ψij, where Ψi,j=∑l=0iψlj(6)
Φ∈b∞r,tp,q:cs if and only if (35) and (44) hold, and (45) also holds with Ψi,j instead of ψij, where Ψi,j=∑l=0iψlj(7)
Φ∈b∞r,tp,q:cs0 if and only if (35) and (44) hold, and (46) also holds with Ψi,j instead of ψij, where Ψi,j=∑l=0iψlj
We recall a basic lemma due to Basar and Altay [47] that will help in characterizing certain classes of matrix mappings from the spaces bsr,tp,q and b∞r,tp,q to any arbitrary space μ.Lemma 19 (see [47]).
Letλ and μ be any two sequence spaces, Φ be an infinite matrix and Ω be a triangular matrix. Then, Φ∈λ:μΩ if and only if ΩΦ∈λ:μ.Now, by combining Lemma19 with Corollaries 16, 17, and 18, we derive the following classes of matrix mappings:Corollary 20.
LetΦ=ϕij be an infinite matrix and define the matrix Cα=cijα by
(47)cijα=∑l=0iα+i−l−1i−lα+iiϕlj,α>1,for all i,j∈ℕ0. Then, the necessary and sufficient conditions that Φ belongs to any one of the classes b1r,tp,q:Csα,b1r,tp,q:C∞α,bsr,tp,q:C∞α,b∞r,tp,q:Csα, and b∞r,tp,q:C∞α can be obtained from the respective ones in Corollaries 16, 17, and 18, by replacing the entries of the matrix Φ by those of matrix Cα, where Csα and C∞α are generalized Cesàro sequence spaces of order α defined by Roopaei et al. [48].Corollary 21.
LetΦ=ϕij be an infinite matrix and define the matrix C~=Cijby(48)C~ij=∑l=0iClCi−lCi+1ϕlj,i,j∈ℕ0,where Ci is the sequence of Catalan numbers. Then, the necessary and sufficient conditions that Φ belongs to any one of the classes b1r,tp,q:c0C~,b1r,tp,q:cC~,bsr,tp,q:c0C~,bsr,tp,q:cC~,b∞r,tp,q:c0C~, and b∞r,tp,q:cC~ can be obtained from the respective ones in Corollaries 16, 17, and 18, by replacing the entries of the matrix Φ by those of matrix C~, where cC~ and c0C~ are Catalan sequence spaces defined by İlkhan [49].Corollary 22.
LetΦ=ϕij be an infinite matrix and define the matrix Cq=cijq by
(49)cijq=∑l=0iqli+1qϕlj,i,j∈ℕ0,where iq is the q-analogue of i∈ℕ0. Then, the necessary and sufficient conditions that Φ belongs to any one of the classes b1r,tp,q:Xsq,b1r,tp,q:X∞q,bsr,tp,q:Xsq,b∞r,tp,q:X∞q,b∞r,tp,q:Xsq, and b∞r,tp,q:X∞q can be obtained from the respective ones in Corollaries 16, 17, and 18, by replacing the entries of the matrix Φ by those of matrix Cq, where Xsq and X∞q are q-Cesàro sequence spaces defined by Yaying et al. [28].


## 5. Geometric Properties

In this section, we examine some geometric properties of the spacebsr,tp,q. Before proceeding, we recall some notions in Banach spaces which are necessary for this investigation. We use the notation Bλ for unit ball in λ.Definition 23 (see [50]).
A Banach spaceλ has the weak Banach-Saks property if every weakly null sequence xi in λ has a subsequence xij whose Cesàro means sequence is norm convergent to zero, that is,
(50)limi⟶∞1i+1∑j=0ixij=0.
Further,λ has the Banach-Saks property if every bounded sequence in λ has a subsequence whose Cesàro means sequence is norm convergent.Definition 24 (see [51]).
A Banach spaceλ has the Banach-Saks type s, if every weakly null sequence xi has a subsequence xij such that, for some K>0,(51)∑j=0ixij≤Ki+11/s,for all i∈ℕ0.Theorem 25.
The sequence spacebsr,tp,q is of Banach-Saks type s.Proof.
Letςi be a sequence of positive numbers satisfying ∑i=0∞ςi≤1/2. Let xi be a weakly null sequence in Bbsr,tp,q. We set z0=x0=0 and z1=xi1=x1. Then, there exists u1∈ℕ0 such that
(52)∑j=u1+1∞z1jejbsr,tp,q<ς1.
Sincexi is a weakly null sequence, we realise that xi⟶0 coordinatewise. Thus, there exists an i2∈ℕ0 such that
(53)∑j=0u1xijejbsr,tp,q<ς1,when i≥i2. We again set z2=xi2. Then, there exists u2>u1 such that
(54)∑j=u2+1∞z2jejbsr,tp,q<ς2.
We again use the fact thatxi⟶0 coordinatewise, which implies that there exists i3>i2 such that
(55)∑j=0u2xijejbsr,tp,q<ς2,when i≥i3.
Continuing this process will lead us to two increasing sequencesij and uj such that
(56)∑j=0ulxijejbsr,tp,q<ςl,for all i≥ij+1 and
(57)∑j=ul+1∞zljejbsr,tp,q<ςl.where zl=xil. Thus
(58)∑l=0izlbsr,tp,q=∑l=0i∑j=0ul−1zljej+∑j=ul−1+1ulzljej+∑j=ul∞zljejbsr,tp,q≤∑l=0i∑j=ul−1+1ulzljejbsr,tp,q+2∑l=0iςl.
Now, sincexi∈Bbsr,tp,q and xbsr,tp,q=∑i=0∞∑j=0ibijr,tp,qxj, we realise that xbsr,tp,q≤1. Therefore, we have
(59)∑l=0i∑j=ul−1+1ulzljejbsr,tp,qp=∑l=0i∑j=ul−1+1ul∑m=0jbjmr,tp,qzlmp≤∑l=0i∑j=0∞∑m=0jbjmr,tp,qzlmp≤i+1.
Now using the fact that1≤i+11/s for all i∈ℕ0 and 1≤s<∞, we obtain
(60)∑l=0izlbsr,tp,q≤i+11/s+1≤2i+11/s.
Thus, we conclude thatbsr,tp,q is of the Banach-Saks type s.☐Definition 26.
The Gurarii’s modulus of convexity of a normed linear spaceλ is defined by
(61)βλς=inf1−inf0≤ι≤1ιx+1−ιy:x,y∈Bλ,x−y=ς,where0≤ς≤2.Theorem 27.
The Gurarii’s modulus of convexity of the normed spacebsr,tp,q is
(62)βbsr,tp,q≤1−1−ς2s1/s,where0≤ς≤2.Proof.
Letx∈bsr,tp,q. Then
(63)xbsr,tp,q=Br,tp,qxℓs=∑i=0∞Br,tp,qxis1/s.
Let0≤ς≤2 and consider the following two sequences:
(64)x=Cr,tp,q1−ς2s1/s,Cr,tp,qς2,0,0,⋯,y=Cr,tp,q1−ς2s1/s,Cr,tp,q−ς2,0,0,⋯,where the matrix Cr,tp,q=cijr,tp,q is the inverse of the matrix Br,tp,q. Then, we observe that
(65)xbsr,tp,q=Br,tp,qxℓs=1−ς2s1/ss+ς2s=1−ς2s+ς2s=1,ybsr,tp,q=Br,tp,qyℓs=1−ς2s1/ss+−ς2s=1−ς2s+ς2s=1.x−ybsr,tp,q=Br,tp,qx−Br,tp,qyℓs=1−ς2s1/s−1−ς2s1/ss+ς2−−ς2s1/s=ς.
Finally, for0≤ι≤1, we have
(66)inf0≤ι≤1ιx+1−ιybsr,tp,q=inf0≤ι≤1ιBr,tp,qx+1−ιBr,tp,qyℓs=inf0≤ι≤1ι1−ς2s1/s+1−ι1−ς2s1/ss+ις2+1−ι−ς21/s=inf0≤ι≤11−ς2s+2ι−1sς2s1/s=1−ς2s1/s.
Consequently,βbsr,tp,qς≤1−1−ς/2s1/s. This completes the proof.☐Corollary 28.
The following results hold:(1)
Ifς=2, then βbsr,tp,qς≤1. Hence, bsr,tp,q is strictly convex(2)
If0<ς<2, then 0<βbsr,tp,qς<1. Hence, bsr,tp,q is uniformly convex


## 6. Conclusion

Thep,q-Euler matrix Br,t of order r,t generalizes some of the well-known matrices presented in the literature, for instance, Binomial matrix Br,t of order r,t [39, 40], Euler matrix of order r [10, 11], etc. Thus, the results presented in this paper strengthen the results of [11, 14, 40, 52–55]. As for future scope, we shall study the domain of the matrix Br,tp,q in the spaces c and c0 of convergent and null sequences, respectively.



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