REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, cilt.113, sa.1, ss.281-297, 2019 (SCI-Expanded)
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, 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 . We also give some remarks, observations and computations for sums of infinite series including these interpolation functions.