$\alpha$, $\beta$-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.