Bi-infinite Riordan matrices: a matricial approach to multiplication and composition of Laurent series
Luis Felipe Prieto-Martínez, Javier Rico
TL;DR
This work develops a bi-infinite matrix framework to generalize Riordan matrices to formal Laurent series, enabling simultaneous treatment of multiplication and composition through bi-infinite matrices. It extends the First Fundamental Theorem for Riordan Matrices to generalized Toeplitz and Lagrange subgroups and constructs a generalized bi-infinite Riordan group $\bm R_{\alpha,\omega}$ with explicit product and inversion rules. The approach requires precise finiteness conditions, notably order constraints on Laurent series, to ensure well-defined products and inverses. As an application, it derives a palindromic (Dehn–Sommerville) identity for $f$- and $h$-polynomials of simplicial complexes, illustrating the framework's ability to obtain combinatorial results beyond the classical Riordan setting and suggesting several avenues for further theoretical and applied development.
Abstract
We propose and investigate a bi-infinite matrix approach to the multiplication and composition of formal Laurent series. We generalize the concept of Riordan matrix to this bi-infinite context, obtaining matrices that are not necessarily lower triangular and are determined, not by a pair of formal power series, but by a pair of Laurent series. We extend the First Fundamental Theorem of Riordan Matrices to this setting, as well as the Toeplitz and Lagrange subgroups, that are subgroups of the classical Riordan group. Finally, as an illustrative example, we apply our approach to derive a classical combinatorial identity that cannot be proved using the techniques related to the classical Riordan group, showing that our generalization is not fruitless.