On the Square Root of a Bell Matrix

Abstract In the context of Riordan arrays, problem determining square root a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = ( f t ) / , defined by formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ ∑ k ≥ 0 with $$f(0)=f_0=0$$ is presented. It proved that if $$f^\prime (0)=1$$ ′ 1 and $$f^{\prime \prime }(0)\ne 0$$ ″ ≠ then there exists another $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H h such $$H*H=R;$$ ∗ ; in particular, function h ( t ) univocally determined symbolic computational method which many situations allows to find closed form. Moreover, it shown related solution Schröder’s equation. We also compute involution this kind matrices.