On the Square Root of a Bell Matrix

In the context of Riordan arrays, the problem of determining the square root of a Bell matrix $R=Ra(f(t)/t, f(t))$ defined by a formal power series $f(t)=sum_{k geq 0}f_kt^k$ with $f(0)=f_0=0$ is presented. It is proved that if $f^prime(0)=1$ and $f^{primeprime}(0) eq 0$ then there exists another Bell matrix $H=Ra(h(t)/t, h(t))$ such that $Hast H=R;$ in particular, function $h(t)$ is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function $h(t)$ is related to the solution of Schr"oder's equation. We also compute a Riordan involution related to this kind of matrices.