Combinatorial sums and generation algorithms for beta-based words and polynomials

Abstract In this paper, by virtue of the beta polynomials, we introduce and study a new special word family referred to as the beta-based words. In order to construct the beta-based words and their generation algorithm, we inspire from the definition of the beta polynomials. By making implementation in the Wolfram language, we obtain some tables for these words and their decimal equivalents. We also show that the decimal equivalents of the beta-based words are associated with the Mersenne numbers. For their visual illustrations, we also present plots designed by black-white superimposed square blocks emerging from the first few beta-based words. Moreover, we investigate some properties of the beta-based words, involving length, height, slope, symmetry and complement properties. Additionally, in this paper, we construct a generalization of harmonic numbers. We also derive some combinatorial sums related to the beta polynomials, the Bernstein polynomials and the generalized harmonic numbers.