On interpolation functions for the number of k-ary Lyndon words associated with the Apostol–Euler numbers and their applications

The aim of this paper is to construct interpolation functions for the numbers of the k-ary Lyndon words which count n digit primitive necklace class representative on the set of the k-letter alphabet. By using the unified zeta-type function and the unification of the Apostol-type numbers which are defined by Ozden et al. (Comput Math Appl 60:2779–2787, 2010), we give an alternating series for the numbers of the k-ary Lyndon words, Lkn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{k}\left( n\right) $$\end{document} in terms of the Apostol–Euler numbers and Frobenius–Euler numbers. We investigate various properties of these functions. Furthermore, applying higher order derivative operator to the interpolation functions for the Lyndon words, we derive ODEs including Stirling-type numbers, the Apostol–Euler numbers, the unified zeta-type functions and also combinatorial sums. By using recurrence relation of the Apostol–Euler numbers, we give computation algorithms for computing not only the Apostol–Euler numbers but also the interpolation functions of the numbers Lkn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{k}\left( n\right) $$\end{document}. We also give some remarks, observations and computations for sums of infinite series including these interpolation functions.