Properties of m-Bonacci Words

The m-Bonacci word is the unique fixed point of the morphism \(\varphi _m:\) \(0\rightarrow 01,~1\rightarrow 02,~2\rightarrow 03,\ldots ,(m-2)\rightarrow 0(m-1),~(m-1)\rightarrow 0\) . The finite m-Bonacci word \(w_{n,m}\) is defined as \(w_{n,m}=\varphi _m^n(0)\) . We study some combinatorial properties of finite m-Bonacci words. We find the values of n, such that \(w_{n,m}\) is square free. We prove that \(w_{n,m}\) is primitive and has a unique representation as a product of two palindromes. We also show that the language \(W_m^0=\{w_{n,m}:~n\ge 0\}\) is context-free free and not dense.