Smooth words on 2-letter alphabets having same parity

In this paper, we consider smooth words over 2-letter alphabets , where are integers having same parity, with . We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either and are odd, or are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for with odd, the frequency of ’s is . [Copyright &y& Elsevier]