For integers $m>r\ge 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d,h)={\left(d_{n,k}\right)}_{n,k\in \mathbb{N}}$ as $d_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots$, and the $(m,r)$-central coefficient triangle of $G$ as $$G^{(m,r)}={\left(d_{mn+r,(m-1)n+k+r}\right)}_{n,k\in \mathbb{N}}.$$ It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h\left(0\right)=0$ and $d\left(0\right),h^{\text{'}}\left(0\right)\ne 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^{(m,r)}$ in terms of the Riordan array $G$. Meanwhile, the...

We give analogs of the complexity $p\left(n\right)$ and of Sturmian words which are called respectively the $*$-complexity $p_{*}\left(n\right)$ and $*$-Sturmian words. We show that the class of $*$-Sturmian words coincides with the class of words satisfying $p_{*}\left(n\right)\le n+1$, and we determine the structure of $*$-Sturmian words. For a class of words satisfying $p_{*}\left(n\right)=n+1$, we give a general formula and an upper bound for $p\left(n\right)$. Using this general formula, we give explicit formulae for $p\left(n\right)$ for some words belonging to this class. In general, $p\left(n\right)$ can take large values, namely,...