$B$-expansion of pseudo-involution in the Riordan group

Abstract: Each numerical sequence $\left( {{b}{0}},{{b}{1}},{{b}{2}},... \right)$ with the generating function $B\left( x \right)$ defines the pseudo-involution in the Riordan group $\left( 1,xg\left( x \right) \right)$ such that $g\left( x \right)=1+xg\left( x \right)B\left( {{x}^{2}}g\left( x \right) \right)$. In the present paper we realize a simple idea: express the coefficients of the series ${{g}^{m}}\left( x \right)$ in terms of the coefficients of the series $B\left( x \right)$. Obtained expansion has a bright combinatorial character, sheds light on the connection of the pseudo-involution in the Riordan group with the generalized binomial series, and is also useful for finding the series $g\left( x \right)$ by the given series $B\left( x \right)$. We compare this expansion with the similar expansion for the sequence $\left( 1,{{a}{1}},{{a}_{2}},... \right)$ with the generating function $A\left( x \right)$ such that $g\left( x \right)=A\left( xg\left( x \right) \right)$.