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

Kucukoglu I., ŞİMŞEK Y.

REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, vol.113, no.1, pp.281-297, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 113 Issue: 1
  • Publication Date: 2019
  • Doi Number: 10.1007/s13398-017-0471-y
  • Journal Name: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.281-297
  • Keywords: Lyndon words, Generating functions, Special numbers, Special polynomials, Differential operator, Algorithm, Stirling numbers of the first kind, Apostol-Euler numbers and polynomials, Frobenius-Euler numbers and polynomials, Arithmetical functions, BERNOULLI
  • Akdeniz University Affiliated: Yes

Abstract

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.