Ask for a special function related to the error function

Question

I am wondering whether anyone knows the following integration has a named special function or a reference

$$ F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y $$

for $a,b, z\in \mathbb{R}$. We are in particular interested in the case when $a\ne 0$.

This function is well defined. Actually it is not hard to see that it satisfies the following bound

$$ |F_{a,b}(z)|\le \text{erf}(|z|). $$

Thanks for any reference and help!

Answer 1

For $a=0$ it reduces to Owen's T-function:

$$\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(b y)\: e^{-y^2} \text{d}y=4T\left(\sqrt{2} \,b z,1/b\right)+\text{erf}\,(z) \,\text{erf}\,(b z)-\frac{2}{\pi}\, \text{arccot}\, b$$

(here's an amusing commentary by a Mathematica developer on this somewhat obscure function)