Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
144 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

$α$, $β$-expansions of the Riordan matrices of the associated subgroup (2005.09590v1)

Published 19 May 2020 in math.NT

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.

Summary

We haven't generated a summary for this paper yet.