$B$-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 $g\left(x\right)=A\left(xg\left(x\right)\right)$.