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We define important properties of the group of infinite lower-triangularmatrices. In particular, we define several subgroups of this group includingsubgroups of the set of Riordan matrices. We also present many examples (infact around 20) of such matrices composed of the sequences of importantfamilies of polynomials or numbers as entries of certain lower-triangularinfinite matrices. By applying the well-known matrix operations mostly such asmultiplication and calculating inverse, we see that such representation can beand is a source of getting new, important relationships between these families

Applications of infinite lower triangular matrices and their group structure in combinatorics and the theory of orthogonal polynomials