On composition of muiltivariable formal power series
Motaz Mokatren
TL;DR
The paper advances the theory of multivariable formal power-series by giving a concrete necessary-and-sufficient condition for the existence of the composition $f \circ G$ with $f=\sum_{\alpha} a_{\alpha} X^{\alpha}$ and $G=(g_1,\dots,g_n)$, together with a Generalized Chain Rule for multivariable fps. It proves a multivariable chain rule $D_j(f\circ G)=\sum_{i=1}^n (D_i f\circ G) D_j g_i$ and the Jacobian form $J_{f\circ G}=J_f(G)\cdot J_G$, and discusses the existence of a composition inverse via $\det(J_G(\theta))\neq 0$. The results unify and extend prior one-variable and mixed-variable cases and have implications for Riordan matrices and multivariable substitution in formal power-series arithmetic. The work also highlights subtle distinctions between unit and nonunit fps in inverse problems and points to future directions for extending the Riordan framework to include unit fps.
Abstract
The paper provides a necessary and sufficient condition for the composition of multivariable formal power series and present the Generalized Chain Rule for formal power series of multiple variables.