English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Each numerical sequence $\left( {{b}_{0}},{{b}_{1}},{{b}_{2}},... \right)$with the generating function $B\left( x \right)$ defines the pseudo-involutionin 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 thepresent 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, shedslight on the connection of the pseudo-involution in the Riordan group with thegeneralized 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 expansionwith the similar expansion for the sequence $\left( 1,{{a}_{1}},{{a}_{2}},...\right)$ with the generating function $A\left( x \right)$ such that $g\left( x\right)=A\left( xg\left( x \right) \right)$.

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