  The main aim of this paper is to investigate and introduce relations between
the numbers of k-ary Lyndon words and unified zeta-type functions which was
defined by Ozden et al [15, p. 2785]. Finally, we give some identities on
generating functions for the numbers of k-ary Lyndon words and some special
numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials,
Frobenius-Euler numbers, Euler numbers and Bernoulli numbers.
