*(English)*Zbl 1224.46009

A linear topological space $X$ over the real field $\mathbb{R}$ is said to be a paranormed space if there is a subadditive function $h:X\to \mathbb{R}$ such that $h\left(\theta \right)=0$, $h\left(x\right)=h(-x)$ and the scalar multiplication is continuous, where $\theta $ denotes the zero vector in $X$.

In the paper under review, the authors introduce some new paranormed sequence spaces defined by Euler and difference operators (i.e., the sequence spaces ${e}_{0}^{r}({\Delta},p)$, ${e}_{c}^{r}({\Delta},p)$, ${e}_{\infty}^{r}({\Delta},p)$ with $p={\left({p}_{k}\right)}_{k\in \mathbb{N}}$ a bounded sequence of positive real numbers) and study some properties of these spaces. In particular, the authors give an inclusion relation between these sequence spaces and study their topological structure. Also, the basis and the $\alpha $-, $\beta $-, and $\gamma $-duals of these spaces are given.