Sequence Spaces, Function Spaces and Approximation TheoryView this Special Issue
On Generalized -Euler Matrix and Associated Sequence Spaces
In this study, we introduce new -spaces and derived by the domain of -analogue of the binomial matrix in the spaces and respectively. We study certain topological properties and inclusion relations of these spaces. We obtain a basis for the space and obtain Köthe-Toeplitz duals of the spaces and We characterize certain classes of matrix mappings from the spaces and to space Finally, we investigate certain geometric properties of the space
1. Introduction and Preliminaries
The -calculus has been a wide and interesting area of research in recent times. Several researchers have worked in the field of -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 -analogue of a known mathematical expression, we mean the generalization of that expression using two independent variables and rather than a single variable as in -calculus. If we put in the -analogue of a known mathematical expression, we get -analogue of that expression. Furthermore, when we receive the original expression. Chakrabarti and Jagannathan  introduced -number to generalize several forms of -oscillator algebras. Since then, several researchers used -theory in different fields of mathematics to extend the theory of single parameter -calculus. We strictly refer to [1–8] for studies in -calculus and  in -calculus.
1.1. Notations and Definitions on -Calculus
Definition 1 (see ). Let Then, twin basic number or -number is defined by Clearly, when reduces to its version .
Definition 2 (see ). The -analogue of binomial coefficient or -binomial coefficient is defined by where -factorial of is given by
Lemma 3. The -binomial formula is defined by
1.2. Sequence Spaces
Let denote the set of all real-valued sequences. Any linear subspace of is called sequence space. The following are some sequence spaces which we shall be frequently used throughout this paper: and denotes the space of all bounded series.
Here, denotes the set of all natural numbers including zero. The sequence spaces and are Banach spaces equipped with the norms respectively.
Let and be two sequence spaces and be an infinite matrix of real entries. By we denote the row of the matrix We say that defines a matrix mapping from to if for every where is -transform of the sequence The notation will denote the family of all matrices that map from to
The matrix domain of the matrix in the space 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 [10–15] 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  introduced Euler sequence space and obtained their -, -, -, and continuous duals, and characterized certain class of matrix mappings on the space where denotes the Euler matrix of order and is defined by for all and The Euler matrix is regular for and is invertible with
Altay et al.  introduced the Euler space and obtained certain inclusion relations, Schauder basis and Köthe-Toeplitz duals of the space As a natural continuation of , Mursaleen et al.  characterized various classes of matrix mappings from the space to other spaces and examined certain geometric properties of the space Further, Altay and Polat  introduced Euler difference spaces and where is backward difference operator defined by for all Extending these spaces, Polat and Başar  studied Euler difference spaces , and of order defined as the set of all sequences whose order backward differences are in the spaces , and respectively. Kadak and Baliarsingh  further generalized these spaces by introducing Euler difference spaces , and of fractional order where is the backward fractional difference operator defined by Kara et al.  introduced paranormed Euler space and studied its topological and geometric properties. Aftermore, Karakaya and Polat  studied paranormed Euler difference sequence spaces , and Extending these spaces, Karakaya et al.  studied paranormed Euler backward difference spaces , and of order. Besides, Demiriz and Çakan  introduced paranormed Euler difference spaces and Furthermore, Kirisci  introduced Euler almost null and Euler almost convergent sequence spaces. Later on, Kara and Başar  introduced generalized difference Euler spaces and where is a generalized difference matrix defined by and characterized certain classes of compact operators on the spaces and Meng and Mei  gave a further generalization of  by introducing Euler difference spaces and where the difference operator is defined by where is a fixed sequence of nonzero real numbers. Recently, Bisgin [39, 40] introduced more generalized Euler space by defining binomial spaces , and and is the binomial matrix defined by
Meng and Song  further generalized these spaces by introducing binomial -difference sequence spaces , and Meng and Mei  studied binomial backward difference sequence spaces , and of fractional order . Besides, Yaying and Hazarika  also studied binomial backward difference spaces of fractional order
Demiriz and Sahin  studied the domain of -Cesàro mean in the spaces and . Very recently, Yaying et al.  studied Banach sequence spaces and defined as the domain of -Cesàro mean in the spaces and respectively, and studied associated operator ideals.
Motivated by the above studies, we generalize Euler mean and Binomial mean in the sense of -theory to and study its domain and in the spaces and respectively. We investigate some topological properties and inclusion relations of the spaces and and obtain a basis for the space In Section 3, we obtain the Köthe-Toeplitz duals (-, -, and -duals) of the spaces and . In Section 4, we characterize some matrix mappings from and spaces to space Section 5 is devoted to investigation of certain geometric properties like Banach-Saks of type and modulus of convexity of the space
In the rest of the paper, unless stated otherwise.
2. Generalized Euler Sequence Spaces and
In this section, we introduce sequence spaces and study their topological properties and some inclusion relations, and obtain a basis for the space
Let be nonnegative real numbers and holds, then the generalized -Euler matrix of order is defined by
One can clearly observe that the matrix reduces to the binomial matrix when Thus, generalizes binomial matrix We may call the matrix as the -analogue of the binomial matrix We also realise that when the matrix reduces to its -version with entries if and otherwise. We call as the -analogue of the binomial matrix Moreover, when then the matrix reduces to with entries if and otherwise. The generalized -Euler sequence spaces and are defined by
The above sequence spaces can be redefined in the notation of (7) by
The spaces and reduce to the following classes of spaces in the special cases of and : (1)When the spaces and reduce to -binomial sequence spaces and respectively, which further reduce to binomial sequence spaces and respectively, when as studied by Bigin (2)When and the spaces and reduce to -Euler space and respectively, which further reduce to well known Euler sequence spaces and respectively, when as studied by Altay et al. (3)When the spaces and reduce to -Euler sequence spaces and
Let us define a sequence in terms of sequence by for each The sequence is called -transform of the sequence . Further, on using (16), we write for each
It is known that if is a -space and is a triangle then the domain of the matrix in the space is also a -space equipped with the norm In the light of this, we have the following result.
Theorem 4. The sequence spaces and are -spaces equipped with the norms defined by respectively.
Proof. The proof is a routine exercise and hence omitted.☐
Theorem 5. The sequence spaces and are linearly isomorphic to and respectively.
Proof. We provide the proof for the space Define the mapping by for all It is easy to observe that is linear and one to one. Let and is as defined in (17). Then, we have Thus, and the mapping is onto and norm preserving. Hence, the space is linearly isomorphic to This completes the proof.☐
Theorem 6. The space is not a Hilbert space, except for the case
Proof. Define the sequences and by We realise that and Then Thus, norm violates the parallelogram identity. Hence, is not a Hilbert space, except for the case ☐
Now we give certain inclusion relations related to the spaces and
Theorem 7. The inclusion strictly holds.
Proof. We provide proof of the inclusion Let for Applying Hölder’s inequality, we have
Thus, where provided exists. This yields the fact that Thus, Similarly, we can show that
Now consider the sequence then it is easy to see that Hence, the inclusion is strict. This completes the proof.☐
Theorem 8. The inclusion strictly holds.
Proof. It is known that inclusion holds for and the mapping is isomorphic, therefore, the inclusion holds. To prove the strictness part, we recall that the inclusion strictly holds for We choose and as defined in (17). Then, This implies that Hence, the inclusion is strict.☐
Theorem 9. The inclusion strictly holds.
Proof. The proof is similar to the proof of Theorem 8. To show the strictness part, we consider the sequence Then, it is clear that Hence, the inclusion strictly holds.☐
We recall that domain of a triangle in space has a basis if and only if has a basis. This statement together with Theorem 5 gives us the following result.
Theorem 10. Let for each Define the sequence of elements of the space for every fixed byThen, the sequence forms a basis for the space and every can be uniquely expressed in the form for each
3. Köthe-Toeplitz Duals
In this section, we obtain Köthe-Toeplitz duals (-, -, and -duals) of the spaces and We omit the proofs for cases and as these can be obtained by analogy and provide proofs for only the case in the current section. First, we recall the definitions of Köthe-Toeplitz duals.
Definition 11. The Köthe-Toeplitz duals or , and -duals of subset are defined by respectively.
Quite recently, Talebi  obtained Köthe-Toeplitz duals of the domain of an arbitrary invertible summability matrix in space. We follow his approach to find the Köthe-Toeplitz duals of the spaces and In the rest of the paper, will denote the family of all finite subsets of and is the complement of
Theorem 12. Define the sets and by Then, and
Proof. Let Let and be the -transform of sequence Then, from the equality (17), we have for all where the matrix is defined by Applying Theorem 2.1 of , we immediately obtained that This completes the proof.☐
Theorem 13. Define the sets