A Generalization of the Chebyshev Polynomials and Nonrooted Posets

In this article we give a generalization of the Chebyshev polynomials of the first kind. Then we describe a Mobius function of the generalized subword order over P-s S is an element of N. These results give the affirmative answer for the conjecture proposed in [A. Bjorner and B. Sagan, "Rationality of the Mobius function of the composition poset," Theoretical Computer Science 359, no. 1-3 (2006): 282-98.] and [B. Sagan and V. Vatter, "The Mobius function of the composition poset," Journal of Algebraic Combinatorics 24, no. 2 (2006): 117-36].