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!



Ask for a special function related to the error function
For $a=0$ it reduces to [Owen's T-function:](https://en.wikipedia.org/wiki/Owen%27s_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](http://blog.wolfram.com/2010/10/07/why-you-should-care-about-the-obscure/) by a Mathematica developer on this somewhat obscure function)


