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Brlek et al. (2008) studied smooth infinite words and established someresults on letter frequency, recurrence, reversal and complementation for2-letter alphabets having same parity. In this paper, we explore smoothinfinite words over $n$-letter alphabet $\{a_1,a_2,...,a_n\}$, where$a_1<a_2<...<a_n$ are positive integers and have same remainder when divided by$n$. And let $a_i=n\cdot q_i+r,\;q_i\in N$ for $i=1,2,...,n$, where$r=0,1,2,...,n-1$. We use distinct methods to prove that (1) if $r=0$, theletters frequency of two times differentiable well-proportioned infinite wordsis $1/n$, which suggests that the letter frequency of the generalized Kolakoskisequences is $1/2$ for 2-letter even alphabets; (2) the smooth infinite wordsare recurrent; (3) if $r=0$ or $r>0 \text{ and }n$ is an even number, thegeneralized Kolakoski words are uniformly recurrent for the alphabet $\Sigma_n$with the cyclic order; (4) the factor set of three times differentiableinfinite words is not closed under any nonidentical permutation. Brlek et al.'sresults are only the special cases of our corresponding results.

Smooth infinite words over $n$-letter alphabets having same remainder when divided by $n$