Küçükoğlu, İremŞimşek, Yılmaz2024-08-202024-08-2020191578-73031579-150510.1007/s13398-017-0471-y2-s2.0-85060501057https://doi.org/10.1007/s13398-017-0471-yhttps://hdl.handle.net/20.500.14591/103The 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.eninfo:eu-repo/semantics/closedAccessLyndon wordsGenerating functionsSpecial numbersSpecial polynomialsDifferential operatorAlgorithmStirling numbers of the first kindApostol-Euler numbers and polynomialsFrobenius-Euler numbers and polynomialsArithmetical functionsArticle2971Q1281113WOS:000456622200022Q1