$\alpha$, $\beta$-expansions of the Riordan matrices of the associated subgroup

TL;DR

This paper studies the decomposition of Riordan matrices into infinite products of specific matrices, deriving formulas for series coefficients and introducing related parameterized series families.

Contribution

It introduces novel decompositions of Riordan matrices into infinite products and derives explicit formulas for associated series coefficients.

Findings

Decomposition of Riordan matrices into infinite products of matrices P_k^α.

Formulas expressing series coefficients in terms of expansion parameters.

Introduction of parameterized series families g_α^(t)(x) and g_β^(t)(x).

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.