The powers of smooth words over arbitrary 2-letter alphabets

A. Carpi (1993) and A. Lepistö (1993) 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. This paper establishes the derivative formula (Theorem 10) of the concatenation of two smooth words and power derivative formula of smooth words over arbitrary 2-letter alphabets. And by making use of power derivative formula (Theorem 12), for arbitrary 2-letter alphabet {a, b} with a, b being positive integers and a < b, we prove that smooth words of length larger than or equal to 2 are h(a, b)-power-free, which means that the power-free index of smooth words is δ(a, b) (Theorem 14), where h(a, b) =    b+ 2, a = 1, b = 3 b+4 2 , 2 | b b+5 2 , 2 ∤ b, anda = 1, b 6= 3 b+3 2 , 2 ∤ b, anda ≥ 2 , δ(a, b) = { b+ 2, a = 1, b = 3 b+ 1, or else . Moreover, we give the number γa,b(n) of smooth words of form w n with a and b having the same parity (Theorem 16). That is, γ1,3(n) = { 0, n ≥ 5 ∞, n < 5 , in other cases, γa,b(n) =    0, n > b 2, h(a, b) ≤ n ≤ b ∞, n < h(a, b) . Thus, we obtain unexpectedly that smooth words are quintic-free, and there are infinitely many smooth biquadrates for the alphabet {1, 3}.