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    Öğe
    Analysis of Generating Functions for Special Words and Numbers and Algorithms for Computation
    (Springer Basel Ag, 2022) Kucukoglu, Irem; Milovanovic, Gradimir, V; Simsek, Yilmaz
    Our aim is to construct and compute efficient generating functions enumerating the k-ary Lyndon words having prime number length which arise in many branches of mathematics and computer science. We prove that these generating functions coincide with the Apostol-Bernoulli numbers and their interpolation functions and obtain other forms of these generating functions including not only the Frobenius-Euler numbers, but also the Fubini type numbers. Moreover, we derive some identities, relations and combinatorial sums including the numbers of the k-ary Lyndon words, the Bernoulli numbers and polynomials, the Stirling numbers and falling factorials. Using these generating functions and recurrence relation for the Apostol-Bernoulli numbers, we give two algorithms to compute these generating functions. Using these algorithms, we compute some infinite series formulas including the number of the k-ary Lyndon words on some special classes of primes with the purpose of providing some numerical evaluations about these generating functions. In addition, we approximate these generating functions by the rational functions of the Apostol-Bernoulli numbers to show that the complexity of the aforementioned algorithms may be decreased by means of approximation method which are illustrated by some numerical evaluations with their plots for varying prime numbers. Finally, using Bell polynomials (i.e., exponential functions) approach to the numbers of Lyndon words, we construct the exponential generating functions for the numbers of Lyndon words. Finally, we define a new family of special numbers related to these special words and investigate some of their fundamental properties.
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    Öğe
    Computational identities for extensions of some families of special numbers and polynomials
    (Tubitak Scientific & Technological Research Council Turkey, 2021) Kucukoglu, Irem; Simsek, Yilmaz
    The main purpose of this paper is to obtain computational identities and formulas for a certain class of combinatorial-type numbers and polynomials. By the aid of the generating function technique, we derive a recurrence relation and an infinite series involving the aforementioned class of combinatorial-type numbers. By applying the Riemann integral to the combinatorial-type polynomials with multivariables, we present some integral formulas for these polynomials, including the Bernoulli numbers of the second kind. By the implementation of the p-adic integral approach to the combinatorial-type polynomials with multivariables, we also obtain formulas for the Volkenborn integral and the fermionic p-adic integral of these polynomials. Furthermore, we provide an approximation for the combinatorial-type numbers with the aid of the Stirling's approximation for factorials. By coding some of our results in Mathematica using the Wolfram programming language, we also provide some numerical evaluations and illustrations on the combinatorial type numbers and their Stirling's approximation with table and figures. We also give some remarks and observations on the combinatorial-type numbers together with their relationships to other well-known special numbers and polynomials. As a result of these observations, we derive some computation formulas containing the Dirichlet series involving the Mobius function, the Bernoulli numbers, the Catalan numbers, the Stirling numbers, the Apostol-Bernoulli numbers, the Apostol-Euler numbers, the Apostol-Genocchi numbers and some kinds of combinatorial numbers. Besides, some inequalities for the combinatorial-type numbers are presented. Finally, we conclude this paper by briefly overviewing the results with their potential applications.
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    Öğe
    Construction and computation of unified Stirling-type numbers emerging from p-adic integrals and symmetric polynomials
    (Springer-Verlag Italia Srl, 2021) Kucukoglu, Irem; Simsek, Yilmaz
    The aim of this paper is to give construction and computation methods for generalized and unified representations of Stirling-type numbers and Bernoulli-type numbers and polynomials. Firstly, we define generalized and unified representations of the falling factorials. By using these new representations as components of the generating functions, we also construct generalized and unified representations of Stirling-type numbers. By making use of the symmetric polynomials, we give computational formulas and algorithm for these numbers. Applying Riemann integral to the unified falling factorials, we introduce new families of Bernoulli-type numbers and polynomials of the second kind by their computation formulas and plots drawn by the Wolfram programming language in Mathematica. Applying p-adic integrals to the unified falling factorials, we construct two new sequences that involve some well-known special numbers such as the Stirling numbers, the Bernoulli numbers and the Euler numbers. Finally, we give not only further remarks and observations, but also some open questions regarding the potential applications and relations of our results.
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    Öğe
    Construction of Bernstein-Based Words and Their Patterns
    (Wiley, 2025) Kucukoglu, Irem; Simsek, Yilmaz
    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.