Groups of generalized Pascal matrices*1

Abstract

We present a new approach to the study of generalized Pascal matrices that yields general results about group structures and explicit formulas for powers and inverses of the Pascal matrices, shows how the groups of Pascal matrices are related to groups of 2×2 matrices, and clarifies the way in which symmetries and duality relate to the algebraic aspects of the theory. We consider bilaterally infinite generalized Pascal matrices obtained as the matrix representations of certain linear operators on spaces of formal Laurent series. Their column generating-functions are simple rational functions closely related to the linear fractional maps of the complex plane. We obtain explicit expressions for the inverses and the powers of the Pascal matrices that generalize most of the analogous results in the literature.