On Generalized <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 37.5255 16.7875" width="37.5255pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g><g transform="matrix(.017,0,0,-0.017,16.115,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.903,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g><g transform="matrix(.017,0,0,-0.017,31.33,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Euler Matrix and Associated Sequence Spaces

Yaying, Taja; Hazarika, Bipan; Mursaleen, Mohammad

doi:https://doi.org/10.1155/2021/8899960

Journal of Function Spaces

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Sequence Spaces, Function Spaces and Approximation Theory

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Research Article | Open Access

Volume 2021 | Article ID 8899960 | https://doi.org/10.1155/2021/8899960

On Generalized -Euler Matrix and Associated Sequence Spaces

Taja Yaying,¹Bipan Hazarika,²and Mohammad Mursaleen^3,4

Academic Editor: Mitsuru Sugimoto

Received19 Sept 2020

Accepted26 May 2021

Published21 Jun 2021

Abstract

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 [1] 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 [9] in -calculus.

1.1. Notations and Definitions on -Calculus

Definition 1 (see [5]). Let Then, twin basic number or -number is defined by Clearly, when reduces to its version .

Definition 2 (see [4]). 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 [10] 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. [11] introduced the Euler space and obtained certain inclusion relations, Schauder basis and Köthe-Toeplitz duals of the space As a natural continuation of [11], Mursaleen et al. [14] characterized various classes of matrix mappings from the space to other spaces and examined certain geometric properties of the space Further, Altay and Polat [29] introduced Euler difference spaces and where is backward difference operator defined by for all Extending these spaces, Polat and Başar [30] 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 [31] 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. [32] introduced paranormed Euler space and studied its topological and geometric properties. Aftermore, Karakaya and Polat [33] studied paranormed Euler difference sequence spaces , and Extending these spaces, Karakaya et al. [34] studied paranormed Euler backward difference spaces , and of order. Besides, Demiriz and Çakan [35] introduced paranormed Euler difference spaces and Furthermore, Kirisci [36] introduced Euler almost null and Euler almost convergent sequence spaces. Later on, Kara and Başar [37] 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 [38] gave a further generalization of [37] 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 [41] further generalized these spaces by introducing binomial -difference sequence spaces , and Meng and Mei [42] studied binomial backward difference sequence spaces , and of fractional order . Besides, Yaying and Hazarika [27] also studied binomial backward difference spaces of fractional order

For the -Cesàro matrix [43, 44] is defined by

Demiriz and Sahin [45] studied the domain of -Cesàro mean in the spaces and . Very recently, Yaying et al. [28] 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 [40](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. [11](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 [25] 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 [25], we immediately obtained that This completes the proof.☐

Theorem 13. Define the sets , and by Then, and

Proof. Let and be the -transform of sequence Then, from the equality (17), we get for each where the matrix is defined by for all
Thus, by applying Theorem 2.2 of [25], we straightly get This completes the proof.☐

Theorem 14. Define the set by Then, and

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 spaces and to space . The following theorem is fundamental in our investigation.

Theorem 15. Let and be an arbitrary subset of Then, if and only if for each and where for all

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) if and only if (2) if and only if (35) and (36) hold, and (37) and also hold(3) if and only if (35) and (36) hold, and (37) and also hold(4) if and only if (35) and (36) hold, and also holds(5) if and only if (35) and (36) hold, and (37) also holds with instead of where (6) if and only if (35) and (36) hold, and (37) and (38) also hold with instead of where (7) if and only if (35) and (36) hold, and (37) and (39) also hold with instead of where

Corollary 17. The following statements hold: (1) if and only if (35) holds, andalso hold (2) if and only if (35) and (41) hold, and (38) and (42) also hold(3) if and only if (35) and (41) hold, (39) and (42) also hold(4) if and only if (35) and (41) hold, andalso holds (5) if and only if (35) and (41) hold, and (42) also holds with instead of where (6) if and only if (35) and (41) hold, and (38) and (42) also hold(7) if and only if (35) and (41) hold, and (39) and (42) also hold with instead of where

Corollary 18. The following statements hold: (1) if and only if (35) and hold, and (42) also holds with (2) if and only if (35) and (44) hold, and (38) and also hold(3) if and only if (35) and (44) hold, and also holds(4) if and only if (35) and (44) hold, and (43) also holds with (5) if and only if (35) and (44) hold, and (42) also hold with and instead of where (6) if and only if (35) and (44) hold, and (45) also holds with instead of where (7) if and only if (35) and (44) hold, and (46) also holds with instead of where We recall a basic lemma due to Baar and Altay [47] that will help in characterizing certain classes of matrix mappings from the spaces and to any arbitrary space

Lemma 19 (see [47]). Let and be any two sequence spaces, be an infinite matrix and be a triangular matrix. Then, if and only if

Now, by combining Lemma 19 with Corollaries 16, 17, and 18, we derive the following classes of matrix mappings:

Corollary 20. Let be an infinite matrix and define the matrix by for all Then, the necessary and sufficient conditions that belongs to any one of the classes , and can be obtained from the respective ones in Corollaries 16, 17, and 18, by replacing the entries of the matrix by those of matrix where and are generalized Cesàro sequence spaces of order defined by Roopaei et al. [48].

Corollary 21. Let be an infinite matrix and define the matrix bywhere is the sequence of Catalan numbers. Then, the necessary and sufficient conditions that belongs to any one of the classes , and can be obtained from the respective ones in Corollaries 16, 17, and 18, by replacing the entries of the matrix by those of matrix where and are Catalan sequence spaces defined by lkhan [49].

Corollary 22. Let be an infinite matrix and define the matrix by where is the -analogue of Then, the necessary and sufficient conditions that belongs to any one of the classes and can be obtained from the respective ones in Corollaries 16, 17, and 18, by replacing the entries of the matrix by those of matrix where and are -Cesàro sequence spaces defined by Yaying et al. [28].

5. Geometric Properties

In this section, we examine some geometric properties of the space Before proceeding, we recall some notions in Banach spaces which are necessary for this investigation. We use the notation for unit ball in

Definition 23 (see [50]). A Banach space has the weak Banach-Saks property if every weakly null sequence in has a subsequence whose Cesàro means sequence is norm convergent to zero, that is, Further, has the Banach-Saks property if every bounded sequence in has a subsequence whose Cesàro means sequence is norm convergent.

Definition 24 (see [51]). A Banach space has the Banach-Saks type if every weakly null sequence has a subsequence such that, for some for all

Theorem 25. The sequence space is of Banach-Saks type

Proof. Let be a sequence of positive numbers satisfying Let be a weakly null sequence in We set and Then, there exists such that Since is a weakly null sequence, we realise that coordinatewise. Thus, there exists an such that when We again set Then, there exists such that We again use the fact that coordinatewise, which implies that there exists such that when
Continuing this process will lead us to two increasing sequences and such that for all and where Thus Now, since and we realise that Therefore, we have Now using the fact that for all and we obtain Thus, we conclude that is of the Banach-Saks type ☐

Definition 26. The Gurarii’s modulus of convexity of a normed linear space is defined by

Theorem 27. The Gurarii’s modulus of convexity of the normed space is

Proof. Let Then Let and consider the following two sequences: where the matrix is the inverse of the matrix Then, we observe that Finally, for we have Consequently, This completes the proof.☐

Corollary 28. The following results hold: (1)If then Hence, is strictly convex(2)If then Hence, is uniformly convex

6. Conclusion

The -Euler matrix of order generalizes some of the well-known matrices presented in the literature, for instance, Binomial matrix of order [39, 40], Euler matrix of order [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 in the spaces and of convergent and null sequences, respectively.

Data Availability

All the data are included within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors are grateful to the anonymous reviewers for careful reading and making necessary corrections which have improved the readability of the paper. The first author (T. Yaying) is thankful to Dr. M.Q. Khan, Principal, Dera Natung Government College, Itanagar, for constant encouragement and administrative supports. The research of the first author (T. Yaying) is supported by SERB, DST (Department of Science and Technology), New Delhi, India, under the grant number EEQ/2019/000082.

References

R. Chakrabarti and R. Jagannathan, “A -oscillator realization of two parameters quantum algebras,” Journal of Physics A: Mathematical and General, vol. 24, no. 13, pp. L711–L718, 1991.
View at: Publisher Site | Google Scholar
S. Araci, U. Duran, M. Acikgoz, and H. M. Srivastava, “A certain -derivative operator and associated divided differences,” Journal of Inequalities and Applications, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
M. Burban and A. U. Kilmyk, “-differentiation, -integration and -hypergeometric functions related to quantum groups,” Integral Transforms and Special Functions, vol. 2, no. 1, pp. 15–36, 1994.
View at: Publisher Site | Google Scholar
R. B. Corcino, “On -binomial coefficients,” Integers: Electronic Journal of Combinatorial Number Theory, vol. 8, p. A29, 2008.
View at: Google Scholar
R. Jagannathan and K. S. Rao, “Two-parameter quantum algebras, twin basic numbers, and associated generalized hypergeometric series,” in Proceedings of the international Conference on Number Theory and Mathematical Physics, Srinivasa Ramanujan Centre, pp. 20-21, Kumbakonam, India, December 2005.
View at: Google Scholar
M. Mursaleen, K. J. Ansari, and A. Khan, “On -analogue of Bernstein operators,” Applied Mathematics and Computation, vol. 266, pp. 874–882, 2015.
View at: Publisher Site | Google Scholar
P. N. Sadjang, “On the fundamental theorem of -calculus and some -Taylor formulas,” Results in Mathematics, vol. 73, p. 39, 2018.
View at: Google Scholar
P. N. Sadjang, “On two -analogues of Laplace transform,” Journal of Difference Equations and Applications, vol. 23, no. 9, pp. 1562–1583, 2016.
View at: Google Scholar
V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.
B. Altay and F. Başar, “On some Euler sequence spaces of non-absolute type,” Ukrainian Mathematical Journal, vol. 57, no. 1, pp. 1–17, 2005.
View at: Publisher Site | Google Scholar
B. Altay, F. Başar, and M. Mursaleen, “On the Euler sequence spaces which include the spaces and I,” Information Sciences, vol. 176, no. 10, pp. 1450–1462, 2006.
View at: Publisher Site | Google Scholar
E. E. Kara and M. Başarır, “An application of Fibonacci numbers into infinite Toeplitz matrices,” Caspian Journal of Mathematical Sciences, vol. 1, no. 1, pp. 43–47, 2012.
View at: Google Scholar
M. Kirisçi and F. Başar, “Some new sequence spaces derived by the domain of generalized difference matrix,” Computers & Mathematcs with Applications, vol. 60, no. 5, pp. 1299–1309, 2010.
View at: Publisher Site | Google Scholar
M. Mursaleen, F. Başar, and B. Altay, “On the Euler sequence spaces which include the spaces and II,” Nonlinear Analysis, vol. 65, no. 3, pp. 707–717, 2006.
View at: Publisher Site | Google Scholar
P.-N. Ng and P.-Y. Lee, “Cesàro sequence spaces of non-absolute type,” Polskiego Towarzystwa Matematycznego, vol. 20, no. 2, pp. 429–433, 1978.
View at: Google Scholar
F. Başar and R. Çolak, Summability Theory and its Applications, Bentham Science Publisher, İstanbul, Turkey, 2012.
M. Mursaleen and F. Başar, “Sequence spaces: topic in modern summability theory,” in Series: Mathematics and its Applications, CRC Press, Taylor & Francis Group, Boca Raton, London, New York, 2020.
View at: Google Scholar
A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies, vol. 85, Elsevier, Amsterdam, 1984.
S. Demiriz, M. İlkhan, and E. E. Kara, “Almost convergence and Euler totient matrix,” Annals of Functional Analysis, vol. 11, no. 3, pp. 604–616, 2020.
View at: Publisher Site | Google Scholar
M. İlkhan and E. E. Kara, “A new Banach space defined by Euler totient matrix operator,” Operators and Matrices, vol. 13, no. 2, pp. 527–544, 2019.
View at: Publisher Site | Google Scholar
M. İlkhan, N. Simsek, and E. E. Kara, “A new regular infinite matrix defined by Jordan totient function and its matrix domain in ,” Mathematicsl Methods in the Applied Sciences, vol. 44, no. 9, pp. 7622–7633, 2021.
View at: Publisher Site | Google Scholar
M. İlkhan, “Matrix domain of a regular matrix derived by Euler totient function in the spaces and ,” Mediterranean Journal of Mathematics, vol. 17, no. 1, p. 27, 2020.
View at: Publisher Site | Google Scholar
H. Roopaei, “Norm of Hilbert operator on sequence spaces,” Journal of Inequalities and Applications, vol. 2020, no. 1, 2020.
View at: Publisher Site | Google Scholar
H. Roopaei, “A study on Copson operator and its associated sequence space,” Journal of Inequalities and Applications, vol. 2020, no. 1, 2020.
View at: Publisher Site | Google Scholar
G. Talebi, “On multipliers of matrix domains,” Journal of Inequalities and Applications, vol. 2018, no. 1, 2018.
View at: Publisher Site | Google Scholar
T. Yaying and B. Hazarika, “On sequence spaces defined by the domain of a regular Tribonacci matrix,” Mathematica Slovaca, vol. 70, no. 3, pp. 697–706, 2020.
View at: Publisher Site | Google Scholar
T. Yaying and B. Hazarika, “On sequence spaces generated by binomial difference operator of fractional order,” Mathematica Slovaca, vol. 69, no. 4, pp. 901–918, 2019.
View at: Publisher Site | Google Scholar
T. Yaying, B. Hazarika, and M. Mursaleen, “On sequence space derived by the domain of -Cesàro matrix in space and the associated operator ideal,” Journal of Mathematical Analysis and Applications, vol. 493, no. 1, article 124453, 2021.
View at: Google Scholar
B. Altay and H. Polat, “On some new Euler difference sequence spaces,” Southeast Asian Bulletin of Mathematics, vol. 30, pp. 209–220, 2006.
View at: Google Scholar
H. Polat and F. Başar, “Some Euler spaces of difference sequences of order ,” Acta Mathematica Scientia B, vol. 27, no. 2, pp. 254–266, 2007.
View at: Google Scholar
U. Kadak and P. Baliarsingh, “On certain Euler difference sequence spaces of fractional order and related dual properties,” Journal of Nonlinear Sciences and Applications, vol. 8, no. 6, pp. 997–1004, 2015.
View at: Publisher Site | Google Scholar
E. E. Kara, M. Öztürk, and M. Başarır, “Some topological and geometric properties of generalized Euler sequence space,” Mathematica Slovaca, vol. 60, no. 3, pp. 385–398, 2010.
View at: Google Scholar
V. Karakaya and H. Polat, “Some new paranormed sequence spaces defined by Euler and difference operators,” Acta Scientiarum Mathematicarum (Szeged), vol. 76, pp. 87–100, 2010.
View at: Google Scholar
V. Karakaya, E. Savas, and H. Polat, “Some paranormed Euler sequence spaces of difference sequences of order,” Mathematica Slovaca, vol. 63, no. 4, pp. 849–862, 2013.
View at: Google Scholar
S. Demiriz and C. Çakan, “On some new paranormed Euler sequence spaces and Euler core,” Acta Mathematica Sinica, English Series, vol. 26, no. 7, pp. 1207–1222, 2010.
View at: Publisher Site | Google Scholar
M. Kirisci, “On the spaces of Euler almost null and Euler almost convergent sequences,” Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, vol. 62, no. 2, pp. 85–100, 2013.
View at: Google Scholar
E. E. Kara and M. Başarır, “On compact operators and some Euler -difference sequence spaces,” Journal of Mathematical Analysis and Applications, vol. 379, no. 2, pp. 499–511, 2011.
View at: Publisher Site | Google Scholar
J. Meng and L. Mei, “The matrix domain and the spectra of a generalized difference operator,” Journal of Mathematical Analysis and Applications, vol. 470, no. 2, pp. 1095–1107, 2019.
View at: Publisher Site | Google Scholar
M. C. Bisgin, “The binomial sequence spaces of nonabsolute type,” Journal of Inequalities and Applications, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
M. C. Bisgin, “The binomial sequence spaces which include the spaces and and geometric properties,” Journal of Inequalities and Applications, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
J. Meng and M. Song, “On some binomial -differnce sequence spaces,” Journal of Inequalities and Applications, vol. 2017, no. 1, 2017.
View at: Publisher Site | Google Scholar
J. Meng and L. Mei, “Binomial difference sequence spaces of fractional order,” Journal of Inequalities and Applications, vol. 2018, no. 1, 2018.
View at: Publisher Site | Google Scholar
H. Aktuğlu and S. Bekar, “On -Cesàro matrix and -statistical convergence,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4717–4723, 2011.
View at: Publisher Site | Google Scholar
S. Bekar, “-matrix summability methods,” Applied Mathematics and Comkputer Science, [Ph.D. Thesis], Eastern Meditarranean University, 2010.
View at: Google Scholar
S. Demiriz and A. Sahin, “-Cesàro sequence spaces derived by -analogues,” Advances in Mathematics, vol. 5, no. 2, pp. 97–110, 2016.
View at: Google Scholar
M. Stieglitz and H. Tietz, “Matrixtransformationen von Folgenräumen eine Ergebnisübersicht,” Mathematische Zeitschrift, vol. 154, no. 1, pp. 1–16, 1977.
View at: Publisher Site | Google Scholar
F. Başar and B. Altay, “On the space of sequences of -bounded variation and related matrix mappings,” Ukrainian Mathematical Journal, vol. 55, no. 1, pp. 136–147, 2003.
View at: Publisher Site | Google Scholar
H. Roopaei, D. Foroutannia, M. İlkhan, and E. E. Kara, “Cesàro spaces and norm of operators on these matrix domains,” Mediterranean Journal of Mathematics, vol. 17, no. 4, p. 121, 2020.
View at: Publisher Site | Google Scholar
M. İlkhan, “A new conservative matrix derived by Catalan numbers and its matrix domain in the spaces and ,” Linear and Multilinear Algebra, vol. 68, no. 2, pp. 417–434, 2020.
View at: Publisher Site | Google Scholar
B. Beauzamy, “Banach-Saks properties and spreading models,” Mathematica Scandinavica, vol. 44, pp. 357–384, 1997.
View at: Publisher Site | Google Scholar
H. Knaust, “Orlicz sequence spaces of Banach-Saks type,” Archiv der Mathematik, vol. 59, no. 6, pp. 562–565, 1992.
View at: Publisher Site | Google Scholar
M. İlkhan, “Certain geometric properties and matrix transformations on a newly introduced Banach space,” Fundamental Journal of Mathematics and Applications, vol. 3, no. 1, pp. 45–51, 2020.
View at: Google Scholar
M. İlkhan and E. E. Kara, “Matrix transformations and compact operators on Catalan sequence spaces,” Journal of Mathematical Analysis and Applications, vol. 498, no. 1, p. 124925, 2021.
View at: Publisher Site | Google Scholar
E. E. Kara and M. Başarır, “On the order difference sequence space of generalized weighted mean and compact operators,” Acta Mathematica Scientia, vol. 33, no. 3, pp. 797–813, 2013.
View at: Google Scholar
P. Z. Alp and M. İlkhan, “On the difference sequence space ,” Mathematical Sciences and Applications E-Notes, vol. 7, no. 2, pp. 161–173, 2019.
View at: Google Scholar

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Copyright © 2021 Taja Yaying et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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