Down-up algebras defined over a polynomial base ring $\K[t_{1}, \cdots, t_{n}]$

In this paper, we study a class of down-up algebras $\A$ defined over a polynomial base ring $\K[t_{1}, \cdots, t_{n}]$ and establish several analogous results. We first construct a $\K-$basis for the algebra $\A$. As a result, we prove that the Gelfand-Kirillov dimension of $\A$ is $n+3$ and completely determine the center of $\A$ when $char\K=0$. Then, we prove that the algebra $\A$ is a noetherian domain if and only if $\beta\neq 0$; and $\A$ is Auslander-regular when $\beta \neq 0$. We also prove that the global dimension of $\A$ is $n+3$; and the algebra $\A$ is a prime ring except $\alpha=\beta=\phi=0$. Moreover, we obtain some results on the Krull dimension, isomorphisms, and automorphisms of the algebra $\A$.