Rate of approximation by modified Gamma-Taylor operators

In this paper we consider the following modification of the Gamma operators which were first introduced in [8] (see [17], [18] and [8] respectively)
$$ A_n(f; x)=\int_0^\infty K_n(x, t)f(t)dt $$
where
$$ K_n(x, t)=\frac{(2n+3)!}{n!(n+2)!}\frac{t^nx^{n+3}}{(x+t)^{2n+4}}, \quad x, t\in(0, \infty), $$
and the following modified Gamma-Taylor operators
$$ A_{n,r}(f;x)=\int_0^\infty K_n(x, t)\left(\sum_{i=0}^r\frac{f^{(i)}(t)}{i!}(x-t)^i\right)dt. $$
We establish some approximation properties of these operators. At the end of the paper we also present some graphs allowing to compare the rate of approximation of $f$ by $A_n(f; x)$ and $A_{n,r}(f; x)$ for certain $n$, $r$ and $x$.