k-ary Lyndon Words and Necklaces Arising as Rational Arguments of Hurwitz–Lerch Zeta Function and Apostol–Bernoulli Polynomials

  • Irem KucukogluEmail author
  • Abdelmejid Bayad
  • Yilmaz Simsek


The main motivation of this paper was to give finite and infinite generating functions for the numbers of the k-ary Lyndon words and necklaces. In order to construct our new generating functions, we use two different methods. The first method is related to the derivative operator \(t\frac{d}{\mathrm{d}t}\) and the Stirling numbers of the second kind. On the other hand, the second method is related to the Hurwitz–Lerch zeta function and the Apostol–Bernoulli numbers. Moreover, by using these generating functions, we give some applications for some selected numerical values including different prime numbers factorization, the Stirling numbers and also the Bernoulli numbers and polynomials.


Lyndon words Necklaces Generating function Special numbers and polynomials Bernoulli numbers and polynomials Apostol–Bernoulli numbers and polynomials Stirling numbers of the second kind Arithmetical function Hurwitz–Lerch zeta function 

Mathematics Subject Classification

03D40 05A15 11A25 11B68 11B83 11M35 68R15 


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Irem Kucukoglu
    • 1
    Email author
  • Abdelmejid Bayad
    • 2
  • Yilmaz Simsek
    • 3
  1. 1.Department of Software Engineering, Faculty of Engineering and ArchitectureAntalya Akev UniversityAntalyaTurkey
  2. 2.LAMMEUniv. Evry, Université Paris-SaclyEvry CedexFrance
  3. 3.Department of Mathematics, Faculty of ScienceUniversity of AkdenizAntalyaTurkey

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