I am interested in understanding whether anything is known about recurrence relations between functions defined via convolution with a fixed kernel and with given initial conditions, namely
$$M_{k+1}(z)=\int_{\mathbb{R}}K(z,z') M_k(z') \, dz'\,,\qquad M_0(z)=B(z)\,.$$
In particular I would like to know if there is a way to solve it, that is to find (if any) a more explicit expression for $M_k(z)$ (even in special cases). I also expect that if the kernel is nice enough the sequence could converge to some fixed point $M_\infty(z)$.
I am not even sure about what would be the right keywords to search for, so thanks for any help also in this direction.