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

TL;DR

This paper introduces a new combinatorial expansion for pseudo-involutions in the Riordan group, linking series coefficients with generalized binomial series to facilitate their computation and understanding.

Contribution

It provides a simple, combinatorial expansion expressing coefficients of powers of g(x) in terms of B(x), illuminating connections with binomial series and aiding in series reconstruction.

Findings

Expansion reveals combinatorial structure of pseudo-involutions

Connects pseudo-involutions with generalized binomial series

Facilitates computation of series g(x) from B(x)

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