The powers of smooth words over arbitrary 2-letter alphabets

TL;DR

This paper investigates the power-freeness of smooth words over any 2-letter alphabet, establishing derivative formulas and proving bounds on the powers that smooth words can contain, revealing new structural properties.

Contribution

It introduces derivative formulas for smooth words over arbitrary 2-letter alphabets and proves their (b+1)-power-freeness, extending prior results beyond the {1,2} alphabet.

Findings

Smooth words are (b+1)-power-free for most 2-letter alphabets {a, b} with a<b.

Smooth words are quintic-free over {1, 3}.

There are infinitely many smooth biquadrates over {1, 3}.

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.