We have proposed a $ q $-analogue $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $ of Fibonacci sequence spaces, where $\mathcal{F}(q) = (f^q_{km})$ denotes a $ q $-Fibonacci matrix defined in the following manner:
$ f^q_{km} = \begin{cases} q^{m+1} \frac{f_{m+1}(q)}{f_{k+3}(q) - 1}, & \text{if } 0 \leq m \leq k, \\ 0, & \text{if } m > k, \end{cases} $
for all $ k, m \in \mathbb{Z}^+_0 $, where $(f_k(q))$ denotes a sequence of $ q $-Fibonacci numbers. We developed a Schauder basis and determined several important duals ($ \alpha $-, $ \beta $-, $ \gamma $-) of the aforesaid constructed spaces $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $. Additionally, we examined certain characterization results for the matrix class $(\mathfrak{U}, \mathfrak{V})$, where $\mathfrak{U} \in \{c(\mathcal{F}(q)), c_0(\mathcal{F}(q))\}$ and $\mathfrak{V} \in \{\ell_{\infty}, c, c_0, \ell_1\}$. Essential conditions for the compactness of the matrix operators on the space $ c_0(\mathcal{F}(q)) $ via the Hausdorff measure of noncompactness (Hmnc) were presented.
Citation: Taja Yaying, S. A. Mohiuddine, Jabr Aljedani. Exploring the $ q $-analogue of Fibonacci sequence spaces associated with $ c $ and $ c_0 $[J]. AIMS Mathematics, 2025, 10(1): 634-653. doi: 10.3934/math.2025028
We have proposed a $ q $-analogue $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $ of Fibonacci sequence spaces, where $\mathcal{F}(q) = (f^q_{km})$ denotes a $ q $-Fibonacci matrix defined in the following manner:
$ f^q_{km} = \begin{cases} q^{m+1} \frac{f_{m+1}(q)}{f_{k+3}(q) - 1}, & \text{if } 0 \leq m \leq k, \\ 0, & \text{if } m > k, \end{cases} $
for all $ k, m \in \mathbb{Z}^+_0 $, where $(f_k(q))$ denotes a sequence of $ q $-Fibonacci numbers. We developed a Schauder basis and determined several important duals ($ \alpha $-, $ \beta $-, $ \gamma $-) of the aforesaid constructed spaces $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $. Additionally, we examined certain characterization results for the matrix class $(\mathfrak{U}, \mathfrak{V})$, where $\mathfrak{U} \in \{c(\mathcal{F}(q)), c_0(\mathcal{F}(q))\}$ and $\mathfrak{V} \in \{\ell_{\infty}, c, c_0, \ell_1\}$. Essential conditions for the compactness of the matrix operators on the space $ c_0(\mathcal{F}(q)) $ via the Hausdorff measure of noncompactness (Hmnc) were presented.
| [1] |
H. Aktuğlu, Ş. Bekar, On $q$-Cesàro matrix and $q$-statistical convergence, J. Comput. Appl. Math., 235 (2011), 4717–4723. https://doi.org/10.1016/j.cam.2010.08.018 doi: 10.1016/j.cam.2010.08.018
|
| [2] |
A. Alotaibi, T. Yaying, S. A. Mohiuddine, Sequence spaces and spectrum of $q$-difference operator of second order, Symmetry, 14 (2022), 1155. https://doi.org/10.3390/sym14061155 doi: 10.3390/sym14061155
|
| [3] |
G. E. Andrews, Fibonacci numbers and the Rogers-Ramanujan identities, Fibonacci Quart., 42 (2004), 3–19. https://doi.org/10.1080/00150517.2004.12428437 doi: 10.1080/00150517.2004.12428437
|
| [4] |
K. İ. Atabey, M. Çınar, M. Et, $q$-Fibonacci sequence spaces and related matrix transformations, J. Appl. Math. Comput., 69 (2023), 2135–2154. https://doi.org/10.1007/s12190-022-01825-9 doi: 10.1007/s12190-022-01825-9
|
| [5] | F. Başar, Summability theory and its applications, Bentham Science Publisher, 2012. https://doi.org/10.2174/97816080545231120101 |
| [6] | M. Başarır, F. Başar, E. E. Kara, On the spaces of Fibonacci difference absolutely $p$-summable, null and convergent sequences, Sarajevo J. Math., 12 (2016), 167–182. |
| [7] |
S. Debnath, S. Saha, Some newly defined sequence spaces using regular matrix of Fibonacci numbers, AKU J. Sci. Eng., 14 (2014), 011301. https://doi.org/10.5578/fmbd.6907 doi: 10.5578/fmbd.6907
|
| [8] | S. Demiriz, A. Şahin, $q$-Cesàro sequence spaces derived by $q$-analogues, Adv. Math. Sci. J., 5 (2016), 97–110. |
| [9] |
H. B. Ellidokuzoğlu, S. Demiriz, On some generalized $q$-difference sequence spaces, AIMS Mathematics, 8 (2023), 18607–18617. https://doi.org/10.3934/math.2023947 doi: 10.3934/math.2023947
|
| [10] |
S. Ercan, Ç. A. Bektaş, Some topological and geometric properties of a new BK-space derived by using regular matrix of Fibonacci numbers, Linear Multilinear A., 65 (2017), 909–921. https://doi.org/10.1080/03081087.2016.1215403 doi: 10.1080/03081087.2016.1215403
|
| [11] | F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203. |
| [12] | A. M. Jarrah, E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat, 17 (2003), 59–78. |
| [13] | V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002. https://doi.org/10.1007/978-1-4613-0071-7 |
| [14] |
E. E. Kara, Some topological and geometric properties of new Banach sequence spaces, J. Inequal. Appl., 2013 (2013), 38. https://doi.org/10.1186/1029-242X-2013-38 doi: 10.1186/1029-242X-2013-38
|
| [15] | E. E. Kara, M. M. Başarır, An application of Fibonacci numbers into infinite Toeplitz matrices, CJMS, 1 (2012), 1–6. |
| [16] | T. Koshy, Fibonacci and Lucas numbers with applications, New York: John Wiley & Sons, Inc., 2001. https://doi.org/10.1002/9781118033067 |
| [17] | B. de Malafosse, The Banach algebra $\mathcal{B}(X)$, where $X$ is a BK space and applications, Mat. Vesnik, 57 (2005), 41–60. |
| [18] | E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik Radova, 9 (2000), 143–234. |
| [19] |
M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal.-Theor., 73 (2010), 2541–2557. https://doi.org/10.1016/j.na.2010.06.030 doi: 10.1016/j.na.2010.06.030
|
| [20] |
M. Mursaleen, A. K. Noman, Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means, Comput. Math. Appl., 60 (2010), 1245–1258. https://doi.org/10.1016/j.camwa.2010.06.005 doi: 10.1016/j.camwa.2010.06.005
|
| [21] | M. Mursaleen, F. Başar, Sequence spaces: Topic in modern summability theory, Boca Raton: CRC Press, 2020. https://doi.org/10.1201/9781003015116 |
| [22] |
M. Mursaleen, S. Tabassum, R. Fatma, On $q$-statistical summability method and its properties, Iran. J. Sci. Technol. Trans. Sci., 46 (2022), 455–460. https://doi.org/10.1007/s40995-022-01285-7 doi: 10.1007/s40995-022-01285-7
|
| [23] |
K. Raj, S. A. Mohiuddine, S. Jasrotia, Characterization of summing operators in multiplier spaces of deferred Nörlund summability, Positivity, 27 (2023), 9. https://doi.org/10.1007/s11117-022-00961-7 doi: 10.1007/s11117-022-00961-7
|
| [24] | I. J. Schur, Ein beitrag zur additiven zahlentheorie, S. B. Akad. Wiss. Berlin, (1917), 302–321. |
| [25] | M. Şengönül, F. Başar, Some new Cesàro sequence spaces of non-absolute type which include the spaces $c_0$ and $c$, Soochow J. Math., 31 (2005), 107–119. |
| [26] |
M. Stieglitz, H. Tietz, Matrixtransformationen von Folgenräumen eine Ergebnisübersicht, Math. Z., 154 (1977), 1–16. https://doi.org/10.1007/BF01215107 doi: 10.1007/BF01215107
|
| [27] |
T. Yaying, F. Başar, Matrix transformation and compactness on $q$-Catalan sequence spaces, Numer. Funct. Anal. Optim., 45 (2024), 373-393. https://doi.org/10.1080/01630563.2024.2349003 doi: 10.1080/01630563.2024.2349003
|
| [28] | T. Yaying, B. Hazarika, F. Başar, On some new sequence spaces defined by $q$-Pascal matrix, T. A. Razmadze Math. In., 176 (2022), 99–113. |
| [29] |
T. Yaying, B. Hazarika, L. Mei, On the domain of $q$-Euler matrix in $c$ and $c_{0}$ with its point spectra, Filomat, 37 (2023), 643–660. https://doi.org/10.2298/FIL2302643Y doi: 10.2298/FIL2302643Y
|
| [30] |
T. Yaying, B. Hazarika, S. A. Mohiuddine, M. Et, On sequence spaces due to $l$th order $q$-difference operator and its spectrum, Iran. J. Sci., 47 (2023), 1271–1281. https://doi.org/10.1007/s40995-023-01487-7 doi: 10.1007/s40995-023-01487-7
|
| [31] |
T. Yaying, B. Hazarika, M. Mursaleen, On sequence space derived by the domain of $q$-Cesàro matrix in $\ell_p$ space and the associated operator ideal, J. Math. Anal. Appl., 493 (2021), 124453. https://doi.org/10.1016/j.jmaa.2020.124453 doi: 10.1016/j.jmaa.2020.124453
|
| [32] |
T. Yaying, B. Hazarika, M. Mursaleen, On generalized $(p, q)$-Euler matrix and associated sequence spaces, J. Funct. Spaces, 2021 (2021), 8899960. https://doi.org/10.1155/2021/8899960 doi: 10.1155/2021/8899960
|
| [33] |
T. Yaying, B. Hazarika, B. C. Tripathy, M. Mursaleen, The spectrum of second order quantum difference operator, Symmetry, 14 (2022), 557. https://doi.org/10.3390/sym14030557 doi: 10.3390/sym14030557
|
| [34] |
T. Yaying, M. İ. Kara, B. Hazarika, E. E. Kara, A study on $q$-analogue of Catalan sequence spaces, Filomat, 37 (2023), 839–850. https://doi.org/10.2298/FIL2303839Y doi: 10.2298/FIL2303839Y
|
| [35] |
T. Yaying, E. Şavas, M. Mursaleen, A novel study on $q$-Fibonacci sequence spaces and their geometric properties, Iran. J. Sci., 48 (2024), 939–951. https://doi.org/10.1007/s40995-024-01644-6 doi: 10.1007/s40995-024-01644-6
|
Taja Yaying, S. A. Mohiuddine, Jabr Aljedani. Exploring the $ q $-analogue of Fibonacci sequence spaces associated with $ c $ and $ c_0 $[J]. AIMS Mathematics, 2025, 10(1): 634-653. doi: 10.3934/math.2025028
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