On non-local approximation properties of the binomial power functions $(1+x^q)^r$

This note mainly concerns the binomial power function, defined as $(1+x^q)^{r}$. We construct systems of polynomials related to non-local approximation, which allows us to establish the density results on $C[a,b]$, where $a,b\in\mathbb{R}$. As a corollary, we show that scattered translated of power functions and certain related functions are dense in the function spaces $L^p([a,b])$, for $1\leq p <\infty$.