Paper
Applications of infinite lower triangular matrices and their group structure in combinatorics and the theory of orthogonal polynomials
Authors
Paweł J. Szabłowski
Abstract
Our focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to each of these operations individually, the set preserves the group structure. The set becomes an algebra with unity when all three operations are considered together. We indicate important properties of the algebraic structures obtained in this way. In particular, we indicate several sub-groups or sub-rings. Among sub-groups, we consider the group of Riordan matrices and indicate its several sub-groups. We show a variety of examples (approximately 20) of matrices that are composed of the sequences of important polynomial or number families as entries of certain lower-triangular infinite matrices. New, significant relationships between these families can be discovered by applying well-known matrix operations like multiplication and inverse calculation to this representation.
The paper intends to compile numerous simple facts about lower-triangular matrices, specifically the family of Rionian matrices, and briefly review their properties.