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