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Analysis of Generating Functions for Special Words and Numbers and Algorithms for Computation

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Abstract

Our aim is to construct and compute efficient generating functions enumerating the k-ary Lyndon words having prime number length which arise in many branches of mathematics and computer science. We prove that these generating functions coincide with the Apostol–Bernoulli numbers and their interpolation functions and obtain other forms of these generating functions including not only the Frobenius–Euler numbers, but also the Fubini type numbers. Moreover, we derive some identities, relations and combinatorial sums including the numbers of the k-ary Lyndon words, the Bernoulli numbers and polynomials, the Stirling numbers and falling factorials. Using these generating functions and recurrence relation for the Apostol–Bernoulli numbers, we give two algorithms to compute these generating functions. Using these algorithms, we compute some infinite series formulas including the number of the k-ary Lyndon words on some special classes of primes with the purpose of providing some numerical evaluations about these generating functions. In addition, we approximate these generating functions by the rational functions of the Apostol–Bernoulli numbers to show that the complexity of the aforementioned algorithms may be decreased by means of approximation method which are illustrated by some numerical evaluations with their plots for varying prime numbers. Finally, using Bell polynomials (i.e., exponential functions) approach to the numbers of Lyndon words, we construct the exponential generating functions for the numbers of Lyndon words. Finally, we define a new family of special numbers related to these special words and investigate some of their fundamental properties.

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Acknowledgements

The research work is supported by the Serbian Academy of Sciences and Arts, \(\Phi \)-96 (G.V. Milovanović) and by the Scientific Research Project Administration of Akdeniz University (Y. Simsek).

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Authors and Affiliations

  1. Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University, 07058, Antalya, Turkey

    Irem Kucukoglu

  2. Serbian Academy of Sciences and Arts, Belgrade, 11000, Serbia

    Gradimir V. Milovanović

  3. Faculty of Sciences and Mathematics, University of Niš, Niš, 18000, Serbia

    Gradimir V. Milovanović

  4. Department of Mathematics, Faculty of Science University of Akdeniz, 07058, Antalya, Turkey

    Yilmaz Simsek

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  1. Irem Kucukoglu
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  2. Gradimir V. Milovanović
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  3. Yilmaz Simsek
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Correspondence to Gradimir V. Milovanović.

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Kucukoglu, I., Milovanović, G.V. & Simsek, Y. Analysis of Generating Functions for Special Words and Numbers and Algorithms for Computation. Mediterr. J. Math. 19, 268 (2022). https://doi.org/10.1007/s00009-022-02191-3

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