Numerical evaluation of special power series including the numbers of Lyndon words: an approach to interpolation functions for Apostol-type numbers and polynomials

Because the Lyndon words and their numbers have practical applications in many different disciplines such as mathematics, probability, statistics, computer programming, algorithms, etc., it is known that not only mathematicians but also statisticians, computer programmers, and other scientists have studied them using different methods. Contrary to other studies, in this paper we use methods associated with zeta-type functions, which interpolate the family of Apostol-type numbers and polynomials of order k. Therefore, the main purpose of this paper is not only to give a special power series including the numbers of Lyndon words and binomial coefficients but also to construct new computational algorithms in order to simulate these series by numerical evaluations and plots. By using these algorithms, we provide novel computational methods to the area of combinatorics on words including Lyndon words. We also define new functions related to these power series, Lyndon words counting numbers, and the Apostol-type numbers and polynomials. Furthermore, we present some illustrations and observations on approximations of functions by rational functions associated with Apostol-type numbers that can provide ideas on the reduction of the algorithmic complexity of these algorithms.