Fractal Generalized Pascal Matrices

The set of generalized Pascal matrices whose entries are generalized binomial coefficients is regarded as a group with respect to Hadamard multiplication. A special system of matrices is introduced and is used to construct fractal generalized Pascal matrices. The Pascal matrix (triangle) is expanded in the Hadamard product of fractal generalized Pascal matrices whose nonzero entries are $p^k$, where $p$ is a fixed prime and $k=0,1,2,\dots$ . The introduced system of matrices suggests the idea of “zero” generalized Pascal matrices, each of which is the limit case of a certain set of generalized Pascal matrices. “Zero” fractal generalized Pascal matrices, which are exemplified by the Pascal triangle modulo 2, are considered.