On Generalized ðp, qÞ-Euler Matrix and Associated Sequence Spaces

In this study, we introduce new BK-spaces br,t s ðp, qÞ and br,t ∞ðp, qÞ derived by the domain of ðp, qÞ-analogue Br,tðp, qÞ 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 br,t s ðp, qÞ and obtain Köthe-Toeplitz duals of the spaces br,t s ðp, qÞ and br,t ∞ðp, qÞ: We characterize certain classes of matrix mappings from the spaces br,t s ðp, qÞ and br,t ∞ðp, qÞ to space μ ∈ fl∞, c, c0, l1, bs, cs, cs0g: Finally, we investigate certain geometric properties of the space br,t s ðp, qÞ:


Introduction and Preliminaries
The ðp, qÞ-calculus has been a wide and interesting area of research in recent times. Several researchers have worked in the field of ðp, qÞ-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 ðp, qÞ-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 ðp, qÞ-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 ðp, qÞ-number to generalize several forms of q-oscillator algebras. Since then, several researchers used ðp, qÞ-theory in different fields of mathematics to extend the theory of single parameter q-calculus. We strictly refer to [1][2][3][4][5][6][7][8] for studies in ðp, qÞ-calculus and [9] in q-calculus.
Definition 2 (see [4]). The ðp, qÞ-analogue of binomial coefficient or ðp, qÞ-binomial coefficient is defined by where ðp, qÞ-factorial ½i pq of i is given by Lemma 3. The ðp, qÞ -binomial formula is defined by 1.2. Sequence Spaces. Let w 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: and bs denotes the space of all bounded series.
Here, ℕ 0 denotes the set of all natural numbers including zero. The sequence spaces ℓ s and ℓ ∞ are Banach spaces equipped with the norms respectively. Let λ and μ be two sequence spaces and Φ = ðϕ ij Þ be an infinite matrix of real entries. By Φ i , we denote the i th row of the matrix Φ: We say that Φ defines a matrix mapping from λ to μ if Φx ∈ μ for every x = ðx j Þ ∈ λ, where Φx = fðΦxÞ i g = f∑ ∞ j=0 ϕ ij x j g is Φ-transform of the sequence x: The notation ðλ : μÞ will denote the family of all matrices that map from λ to μ: The matrix domain λ Φ of the matrix Φ in the space λ is defined by 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][11][12][13][14][15] and textbooks [16][17][18]. For some recent publications dealing with the domain of triangles in classical spaces, we refer [19][20][21][22][23][24][25][26][27][28].
1.3. Literature Review. We give a short survey of literature concerning Euler sequence spaces. Altay and Başar [10] introduced Euler sequence space e r 0 = ðc 0 Þ E r and e r ∞ = ðℓ ∞ Þ E r , obtained their α-, β-, γ-, and continuous duals, and characterized certain class of matrix mappings on the space ðcÞ E r , where E r = ðe r ij Þ denotes the Euler matrix of order r and is defined by for all i, j ∈ ℕ 0 and 0 < r < 1: The Euler matrix E r is regular for 0 < r < 1 and is invertible with ðE r Þ −1 = E 1/r : Altay et al. [11] introduced the Euler space e r s = ðℓ s Þ E r , 1 ≤ s ≤ ∞, and obtained certain inclusion relations, Schauder basis and Köthe-Toeplitz duals of the space e r s : As a natural continuation of [11], Mursaleen et al. [14] characterized various classes of matrix mappings from the space e r s to other spaces and examined certain geometric properties of the space e r s : Further, Altay and Polat [29] introduced Euler difference spaces e r Extending these spaces, Polat and Başar [30] studied Euler difference spaces e r 0 ðΔ Bm Þ,e r c ðΔ Bm Þ, and e r ∞ ðΔ Bm Þ of m th ðm ∈ ℕÞ order defined as the set of all sequences whose m th order backward differences are in the spaces e r 0 ,e r c , and e r ∞ , respectively. Kadak and Baliarsingh [31] where v = ðv j Þ is a fixed sequence of nonzero real numbers. Recently, Bisgin [39,40] introduced more generalized Euler space by defining binomial spaces b r,t s = ðℓ s Þ B r,t , b r,t 0 = ðc 0 Þ B r,t ,b r,t c = ðcÞ B r,t , and b r,t ∞ = ðℓ ∞ Þ B r,t , and B r,t = ðb r,t ij Þ is the binomial matrix defined by Demiriz and Sahin [45] studied the domain of q-Cesàro mean in the spaces c and c 0 . Very recently, Yaying et al. [28] studied Banach sequence spaces X q s and X q ∞ defined as the domain of q-Cesàro mean in the spaces ℓ s and ℓ ∞ , respectively, and studied associated operator ideals.
Motivated by the above studies, we generalize Euler mean E r and Binomial mean B r,t in the sense of ðp, qÞ-theory to B r,t ðp, qÞ and study its domain b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ in the spaces ℓ s and ℓ ∞ , respectively. We investigate some topological properties and inclusion relations of the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ and obtain a basis for the space b r,t s ðp, qÞ: In Section 3, we obtain the Köthe-Toeplitz duals (α-, β-, and γ-duals) of the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ. In Section 4, we characterize some matrix mappings from b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ spaces to space μ ∈ fℓ ∞ , c, c 0 , ℓ 1 , cs, cs 0 , bsg: Section 5 is devoted to investigation of certain geometric properties like Banach-Saks of type s and modulus of convexity of the space b r,t s ðp, qÞ: In the rest of the paper, 1 ≤ s < ∞, unless stated otherwise.

Generalized Euler Sequence Spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ
In this section, we introduce sequence spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ, study their topological properties and some inclusion relations, and obtain a basis for the space b r,t s ðp, qÞ: Let r, t be nonnegative real numbers and 0 < q < p ≤ 1 holds, then the generalized ðp, qÞ-Euler matrix B r,t ðp, qÞ = ðb r,t ij Þ of order ðr, tÞ is defined by One can clearly observe that the matrix B r,t ðp, qÞ reduces to the binomial matrix B r,t when p = q = 1: Thus, B r,t ðp, qÞ generalizes binomial matrix B r,t : We may call the matrix B r,t ðp, qÞ as the ðp, qÞ-analogue of the binomial matrix B r,t : We also realise that when p = 1, the matrix B r,t ðp, qÞ reduces to its q-version B r,t ðqÞ with entries We 3 Journal of Function Spaces call B r,t ðqÞ as the q-analogue of the binomial matrix B r,t : Moreover, when t = 1 − r, then the matrix B r,t ðp, qÞ otherwise. The generalized ðp, qÞ-Euler sequence spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ are defined by The above sequence spaces can be redefined in the notation of (7) by The spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ reduce to the following classes of spaces in the special cases of ðp, qÞ and ðr, tÞ: (1) When p = 1, the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ reduce to q-binomial sequence spaces b r,t s ðqÞ = ðℓ s Þ B r,t ðqÞ and b r,t ∞ ðqÞ = ðℓ ∞ Þ B r,t ðqÞ , respectively, which further reduce to binomial sequence spaces b r,t s and b r,t ∞ , respectively, when q ⟶ 1, as studied by Bisgin [40] (2) When p = 1 and r + t = 1, the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ reduce to q-Euler space e r s ðqÞ = ðℓ s Þ E r ðqÞ and e r ∞ ðqÞ = ðℓ ∞ Þ E r ðqÞ , respectively, which further reduce to well known Euler sequence spaces e r s and e r ∞ , respectively, when q ⟶ 1, as studied by Altay et al. [11] (3) When r + t = 1, the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ reduce to ðp, qÞ-Euler sequence spaces e r s ðp, qÞ = ðℓ s Þ E r ðp,qÞ and e r ∞ ðp, qÞ = ðℓ ∞ Þ E r ðp,qÞ Let us define a sequence y = ðy i Þ in terms of sequence x = ðx j Þ by for each i ∈ ℕ 0 : The sequence y is called B r,t ðp, qÞ-transform of the sequence x. Further, on using (16), we write for each i ∈ ℕ 0 : It is known that if λ is a BK-space and Φ is a triangle then the domain λ Φ of the matrix Φ in the space λ is also a BK-space equipped with the norm kxk λ Φ = kΦxk λ : In the light of this, we have the following result. Theorem 4. The sequence spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ are BK-spaces equipped with the norms defined by respectively.

Journal of Function Spaces
Proof. The proof is a routine exercise and hence omitted. ☐ Theorem 5. The sequence spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ are linearly isomorphic to ℓ s and ℓ ∞ , respectively.
Proof. We provide the proof for the space b r,t s ðp, qÞ: Define the mapping T : b r,t s ðp, qÞ ⟶ ℓ s by Tx = B r,t ðp, qÞx for all x ∈ b r,t s ðp, qÞ: It is easy to observe that T is linear and one to one. Let y = ðy i Þ ∈ ℓ s and x = ðx i Þ is as defined in (17). Then, we have Thus, x ∈ b r,t s ðp, qÞ and the mapping T : b r,t s ðp, qÞ ⟶ ℓ s is onto and norm preserving. Hence, the space b r,t s ðp, qÞ is linearly isomorphic to ℓ s : This completes the proof. ☐ Theorem 6. The space b r,t s ðp, qÞ,1 ≤ s ≤ ∞, is not a Hilbert space, except for the case s = 2: Proof. Define the sequences x = ðx i Þ and y = ðy i Þ by > > > > > > > > > > > > > : We realise that ðB r,t ðp, qÞxÞ i = ð1, 1, 0, 0, ⋯Þ and ðB r,t ðp, qÞyÞ i = ð1,−1, 0, 0, ⋯Þ: Then Thus, b r,t s ðp, qÞ norm violates the parallelogram identity. Proof. We provide proof of the inclusion ℓ s ⊂ b r,t s ðp, qÞ, 1 ≤ s < ∞: Let x = ðx i Þ ∈ ℓ s for 1 < s < ∞: Applying Hölder's inequality, we have Proof. It is known that inclusion ℓ s ⊂ ℓ k holds for 1 ≤ s < k < ∞ and the mapping B r,t ðp, qÞ: b r,t s ðp, qÞ ⟶ ℓ s is isomorphic, therefore, the inclusion b r,t s ðp, qÞ ⊂ b r,t k ðp, qÞ holds. To prove the strictness part, we recall that the inclusion ℓ s ⊂ ℓ k strictly holds for 1 ≤ s < k < ∞: We choose y ∈ ℓ k \ ℓ s and x as defined in (17 Then, the sequence fb ðjÞ ðp, qÞg forms a basis for the space b r,t s ðp, qÞ and every x ∈ b r,t s ðp, qÞ can be uniquely expressed in the form x = ∑ ∞ j=0 ξ j b ðjÞ ðp, qÞ for each j ∈ ℕ 0 :

Köthe-Toeplitz Duals
In this section, we obtain Köthe-Toeplitz duals (α-, β-, and γ-duals) of the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ: 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.
Quite recently, Talebi [25] obtained Köthe-Toeplitz duals of the domain of an arbitrary invertible summability matrix in ℓ s space. We follow his approach to find the Köthe-Toeplitz duals of the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ: In the rest of the paper, N will denote the family of all finite subsets of ℕ 0 and k = s/1 − s is the complement of s: Proof. Let 1 < s < ∞: Let ða i Þ ∈ w and y = ðy i Þ be the B r,t ðp, qÞ -transform of sequence x = ðx i Þ: Then, from the equality (17), we have for all i ∈ ℕ 0 , where the matrix G r,t ðp, qÞ = ðg r,t ij ðp, qÞÞ is defined by Applying Theorem 2.1 of [25], we immediately obtained that and, for each i ∈ ℕ 0 , where the matrix H r,t ðp, qÞ = ðh r,t ij ðp, qÞÞ is defined by

Matrix Mappings
In this section, we characterize a certain class of matrix mappings from the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ to space μ ∈ fℓ ∞ , c, c 0 , ℓ 1 , bs, cs, cs 0 g. The following theorem is fundamental in our investigation.
for all i, j ∈ ℕ 0 : 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: also holds also hold  (35) and hold, and (42) also holds with k = 1 (2) Φ ∈ ðb r,t ∞ ðp, qÞ: cÞ if and only if (35) and (44) hold, and (38) and We recall a basic lemma due to Basar and Altay [47] that will help in characterizing certain classes of matrix mappings from the spaces b r,t s ðp, qÞ and b r,t ∞ ðp, qÞ to any arbitrary space μ: 10 Journal of Function Spaces Lemma 19 (see [47]). Let λ and μ be any two sequence spaces, Φ be an infinite matrix and Ω be a triangular matrix. Then, Φ ∈ ðλ : μ Ω Þ if and only if ΩΦ ∈ ðλ : μÞ: Now, by combining Lemma 19 with Corollaries 16, 17, and 18, we derive the following classes of matrix mappings: Corollary 20. Let Φ = ðϕ ij Þ be an infinite matrix and define the matrix C α = ðc α ij Þ by for all i, j ∈ ℕ 0 : Then, the necessary and sufficient conditions that Φ belongs to any one of the classes ðb r,t 1 ðp, qÞ: C α s Þ, ðb r,t 1 ðp, qÞ: C α ∞ Þ,ðb r,t s ðp, qÞ: C α ∞ Þ,ðb r,t ∞ ðp, qÞ: C α s Þ, and ðb r,t ∞ ðp, qÞ: C α ∞ Þ 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 α , where C α s and C α ∞ are generalized Cesàro sequence spaces of order α defined by Roopaei et al. [48].
where ½i q is the q-analogue of i ∈ ℕ 0 : Then, the necessary and sufficient conditions that Φ belongs to any one of the classes ðb r,t 1 ðp, qÞ: X q s Þ,ðb r,t 1 ðp, qÞ: X q ∞ Þ,ðb r,t s ðp, qÞ: X q s Þ, ðb r,t ∞ ðp, qÞ: X q ∞ Þ,ðb r,t ∞ ðp, qÞ: X q s Þ, and ðb r,t ∞ ðp, qÞ: X q ∞ Þ 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 ðqÞ , where X q s and X q ∞ are q-Cesàro sequence spaces defined by Yaying et al. [28].

Geometric Properties
In this section, we examine some geometric properties of the space b r,t s ðp, qÞ: Before proceeding, we recall some notions in Banach spaces which are necessary for this investigation. We use the notation BðλÞ for unit ball in λ: Definition 23 (see [50]). A Banach space λ has the weak Banach-Saks property if every weakly null sequence ðx i Þ in λ has a subsequence ðx i j Þ whose Cesàro means sequence is norm convergent to zero, that is, 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 space λ has the Banach-Saks type s, if every weakly null sequence ðx i Þ has a subsequence ðx i j Þ such that, for some K > 0, for all i ∈ ℕ 0 : Theorem 25. The sequence space b r,t s ðp, qÞ is of Banach-Saks type s: Proof. Let ðς i Þ be a sequence of positive numbers satisfying ∑ ∞ i=0 ς i ≤ 1/2: Let ðx i Þ be a weakly null sequence in Bðb r,t s ðp, qÞÞ: We set z 0 = x 0 = 0 and z 1 = x i 1 = x 1 : Then, there exists u 1 ∈ ℕ 0 such that Since ðx i Þ is a weakly null sequence, we realise that x i ⟶ 0 coordinatewise. Thus, there exists an i 2 ∈ ℕ 0 such that when i ≥ i 2 : We again set z 2 = x i 2 : Then, there exists u 2 > u 1 such that