Construction of Bernstein‐Based Words and Their Patterns

• Published in: Mathematical methods in the applied sciences Vol. 48; no. 13; pp. 12819 - 12836
• Main Authors: Kucukoglu, Irem, Simsek, Yilmaz
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.09.2025
• Subjects: Basis functions; Bernstein basis functions; combinatorics on words; computational implementation; Farey fractions; Fractions; generating functions; Hypergeometric functions; Polynomials; recurrence relation; special functions; Sums; tree diagrams
• ISSN: 0170-4214; 1099-1476; 1099-1476
• DOI: 10.1002/mma.11066

ABSTRACT In 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.