A generalization of Chebyshev polynomials and non rooted posets

In this paper we give a generalization of Chebyshev polynomials and using this we describe the M"obius function of the generalized subword order from a poset {a1,…as,c |ai<c}, which contains an affirmative answer for the conjecture by Bj"orner, Sagan, Vatter.[5,10]