The powers of smooth words over arbitrary 2-letter alphabets

Abstract

Carpi (1993) and Lepisto (1994) proved independently that smooth words are cube-free for the alphabet {1, 2}, but nothing is known on whether for the other 2-letter alphabets, smooth words are k-power-free for some suitable positive integer k. In this paper, we first establish the derivative formula of the concatenation of two smooth words and power derivative formula of smooth words over arbitrary 2-letter alphabets. Then by making use of power derivative formula, for arbitrary 2-letter alphabet {a, b} with a, b being positive integers and a<b, we prove that smooth words are (b+1)-power-free except for a=1 and b=3; and smooth words are quintic-free and there are infinitely many smooth biquadrates for the alphabet {1, 3}. Moreover, we give the number of smooth words of form w^n with a and b having the same parity.