Construction of Bernstein-Based Words and Their Patterns

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dc.authorid0000-0001-9100-2252
dc.contributor.authorKucukoglu, Irem
dc.contributor.authorSimsek, Yilmaz
dc.date.accessioned2026-01-24T12:30:50Z
dc.date.available2026-01-24T12:30:50Z
dc.date.issued2025
dc.departmentAlanya Alaaddin Keykubat Üniversitesi
dc.description.abstractIn this paper, with inspiration of the definition of Bernstein basis functions and their recurrence relation, we give construction of a new word family that we refer Bernstein-based words. By classifying these special words as the first and second kinds, we investigate their some fundamental properties involving periodicity and symmetricity. Providing schematic algorithms based on tree diagrams, we also illustrate the construction of the Bernstein-based words. For their symbolic computation, we also give computational implementations of Bernstein-based words in the Wolfram Language. By executing these implementations, we present some tables of Bernstein-based words and their decimal equivalents. In addition, we present black-white and four-colored patterns arising from the Bernstein-based words with their potential applications in computational science and engineering. We also give some finite sums and generating functions for the lengths of the Bernstein-based words. We show that these functions are of relationships with the Catalan numbers, the centered m$$ m $$-gonal numbers, the Laguerre polynomials, certain finite sums, and hypergeometric functions. We also raise some open questions and provide some comments on our results. Finally, we investigate relationships between the slopes of the Bernstein-based words and the Farey fractions.
dc.identifier.doi10.1002/mma.11066
dc.identifier.endpage12836
dc.identifier.issn0170-4214
dc.identifier.issn1099-1476
dc.identifier.issue13
dc.identifier.scopus2-s2.0-105009882899
dc.identifier.scopusqualityQ1
dc.identifier.startpage12819
dc.identifier.urihttps://doi.org/10.1002/mma.11066
dc.identifier.urihttps://hdl.handle.net/20.500.12868/5467
dc.identifier.volume48
dc.identifier.wosWOS:001523471500001
dc.identifier.wosqualityQ1
dc.indekslendigikaynakWeb of Science
dc.indekslendigikaynakScopus
dc.language.isoen
dc.publisherWiley
dc.relation.ispartofMathematical Methods in The Applied Sciences
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanı
dc.rightsinfo:eu-repo/semantics/openAccess
dc.snmzKA_WoS_20260121
dc.subjectBernstein basis functions
dc.subjectcombinatorics on words
dc.subjectcomputational implementation
dc.subjectFarey fractions
dc.subjectrecurrence relation
dc.subjectgenerating functions
dc.subjecttree diagrams
dc.subjectspecial functions
dc.titleConstruction of Bernstein-Based Words and Their Patterns
dc.typeArticle

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