A new factorization of the generalized period-doubling sequences through kernel words and gaps sequences

In this paper, we study some new factorizations of period-doubling sequences over a $k$-letter alphabet, where $k\ge 2$. First, we define the combinatorial and arithmetic properties of these sequences. Then, we define the kernel words of period-doubling sequences and demonstrate how to factorize a binary sequence using its kernel words. Next, we define gap sequences for period-doubling sequences and explore their relationship with kernel words. Lastly, we present a factorization of period-doubling sequences for $k\ge 3$ based on kernel words and gap sequences.