Notes on Bounds of the Norms of Various Circulant Matrices with bi-periodic Pell numbers

Notes on Bounds of the Norms of Various Circulant Matrices with bi-periodic Pell numbers | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 22 December 2025 V1 Latest version Share on Notes on Bounds of the Norms of Various Circulant Matrices with bi-periodic Pell numbers Authors : Sukran Uygun 0000-0002-7878-2175 [email protected] and Hulya Aytar Authors Info & Affiliations https://doi.org/10.22541/au.176639164.42949398/v1 109 views 87 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract In this paper, we investigate spectral norm bounds of several classes of circulant-type matrices whose entries are generated by the bi-periodic Pell sequence. This sequence generalizes the classical Pell numbers and establishes connections with bi-periodic Fibonacci and Lucas sequences through two positive real parameters. We first derive new summation formulas for the bi-periodic Pell numbers, which play a key role in the norm analysis. Based on these results, we construct r -circulant, symmetric r-circulant, and geometric circulant matrices associated with the bi-periodic Pell sequence and obtain explicit upper and lower bounds for their spectral norms using different analytical approaches, including Frobenius norm inequalities, matrix norm relations, and entrywise estimates. Furthermore, we establish norm bounds for the Hadamard and Kronecker products of these matrices, providing a unified framework for analyzing matrix products arising from bi-periodic recursive structures. In addition, closed-form expressions for the eigenvalues of the r -circulant matrices are derived, allowing further insight into their spectral properties. The results presented here extend and generalize several existing norm inequalities for circulant matrices in the literature and contribute new analytical tools for the study of structured matrices generated by bi-periodic sequences. Supplementary Material File (boundnorm biperpell.pdf) Download 189.53 KB Information & Authors Information Version history V1 Version 1 22 December 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords circulant matrix matrix norm pell numbers Authors Affiliations Sukran Uygun 0000-0002-7878-2175 [email protected] Gaziantep University View all articles by this author Hulya Aytar Gaziantep University View all articles by this author Metrics & Citations Metrics Article Usage 109 views 87 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Sukran Uygun, Hulya Aytar. Notes on Bounds of the Norms of Various Circulant Matrices with bi-periodic Pell numbers. Authorea . 22 December 2025. DOI: https://doi.org/10.22541/au.176639164.42949398/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. 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