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q-Fibonacci sequence spaces and related matrix transformations

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Abstract

In this paper, we define the sequence spaces \(\ell _{p}(\widehat{F}_{q})\) \(\left( 1\le p< \infty \right) \), \(\ell _{\infty }(\widehat{F}_{q})\), \(c_{0}(\widehat{F}_{q})\) and \(c(\widehat{F}_{q})\) by using q-Fibonacci band matrix \(\widehat{F}_{q}\) defined by

$$\begin{aligned} \widehat{F}_{q}=\widehat{F}_{nk}(q)=\left\{ \begin{array}{lll} -\dfrac{F_{n+1}(q)-1}{q^{n}F_{n}(q)} &{}, k=n-1\\ \dfrac{F_{n+2}(q)-1}{q^{n}F_{n}(q)} &{}, k=n\\ 0 &{}, \text {otherwise} \end{array} \right. \quad (k,n \in \mathbb {N}). \end{aligned}$$

We study some topological properties and give some inclusion relations for these spaces. In addition, we build a bases for the space \(\ell _{p}(\widehat{F}_{q})\), compute \(\alpha \)-, \(\beta \)-, \(\gamma \)- duals of the same space, characterize some matrix classes and examine some geometric properties.

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Acknowledgements

This paper is a part of PhD thesis to be submitted to Fırat University by Koray İbrahim ATABEY. The authors are sincerely grateful to Prof. George E. Andrews for his kind help and encouragement during the preparation of this paper.

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Author notes

  1. Muhammed Çınar and Mikail Et have contributed equally to this work.

Authors and Affiliations

  1. Muş Nizamülmülk Girl Anatolia İmam Hatip High School, 49250, Muş, Turkey

    Koray İbrahim Atabey

  2. Department of Mathematics Education, Muş Alparslan University, 49250, Muş, Turkey

    Muhammed Çınar

  3. Department of Mathematics, Fırat University, 23119, Elazığ, Turkey

    Mikail Et

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  1. Koray İbrahim Atabey
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  2. Muhammed Çınar
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Correspondence to Koray İbrahim Atabey.

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Atabey, K.İ., Çınar, M. & Et, M. q-Fibonacci sequence spaces and related matrix transformations. J. Appl. Math. Comput. 69, 2135–2154 (2023). https://doi.org/10.1007/s12190-022-01825-9

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  • DOI: https://doi.org/10.1007/s12190-022-01825-9

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