Simple proofs of open problems about the structure of involutions in the Riordan group

We prove that if D=(g(x),f(x)) is an element of order 2 in the Riordan group then g(x)=±exp[Φ(x,xf(x)] for some antisymmetric function Φ(x,z). Also we prove that every element of order 2 in the Riordan group can be written as BMB-1 for some element B and M=(1,-1) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596].