$α$, $β$-expansions of the Riordan matrices of the associated subgroup

Abstract

We consider the group of the matrices $\left( 1,g\left( x \right) \right)$ isomorphic to the group of formal power series $g\left( x \right)=x+{{g}_{2}}{{x}^{2}}+...$ under composition: $\left( 1,{{g}_{2}}\left( x \right) \right)\left( 1,{{g}_{1}}\left( x \right) \right)=\left( 1,{{g}_{1}}\left( {{g}_{2}}\left( x \right) \right) \right)$. Denote $P_{k}^{α}=\left( 1,x{{\left( 1-kα{{x}^{k}} \right)}^{{-1}/{k}\;}} \right)$. Matrix $\left( 1,g\left( x \right) \right)$is decomposed into an infinite product of the matrices $P_{k}^{α}$ with suitable exponents in two ways: to left-handed and right-handed products with respect to the matrix $P_{1}^{{{α}_{1}}={{β}_{1}}}$: $\left( 1,g\left( x \right) \right)=...P_{k}^{{{α}_{k}}}...P_{2}^{{{α}_{2}}}P_{1}^{{{α}_{1}}}=P_{1}^{{{β}_{1}}}P_{2}^{{{β}_{2}}}...P_{k}^{{{β}_{k}}}...$. We obtain two formulas expressing the coefficients of the series ${{\left( {g\left( x \right)}/{x}\; \right)}^{z}}$ in terms of the expansion coefficients ${{α}_{i}}$, ${{β}_{i}}$ and introduce two one-parameter families of series $g_{α}^{\left( t \right)}\left( x \right)$ and $g_{β}^{\left( t \right)}\left( x \right)$ associated with these expansions.