Summary:
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 formal 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.
Spanish layman's summary:
Generalizamos el concepto de matrices de Riordan a un entorno bi-infinito adecuado para la multiplicación y composición de series de Laurent formales. Además, ampliamos el Primer Teorema Fundamental de las Matrices de Riordan y mostramos su uso derivando una identidad combinatoria clásica.
English layman's summary:
We generalize the concept of Riordan matrices to a bi-infinite setting suited for the multiplication and composition of formal Laurent Series. We further extend the First Fundamental Theorem of Riordan Matrices and illustrate the framework by deriving a classical combinatorial identity.
Keywords: Formal Laurent series; Bi-infinite matrices; Riordan group
JCR-JIF Impact Factor and WoS quartile: 1,100 - Q2 (2025)
DOI reference:
https://doi.org/10.1016/j.laa.2025.11.010
Published on paper: February 2026.
Published on-line: November 2025.
Citation:
L.F. Prieto-Martínez, J. Rico, "Bi-infinite Riordan matrices: A matricial approach to multiplication and composition of formal Laurent series", Linear Algebra and its Applications, Vol. 731, pp. 139 - 159, February 2026. [Online: November 2025] doi: 10.1016/j.laa.2025.11.010
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