Fractal generalized Pascal matrices

TL;DR

This paper introduces fractal generalized Pascal matrices constructed from a special system of matrices, revealing their structure as Hadamard products and exploring zero matrices like Pascal modulo 2.

Contribution

It defines a new class of fractal generalized Pascal matrices and connects them with zero matrices and modular variants, expanding the understanding of binomial coefficient matrices.

Findings

Pascal matrix is the Hadamard product of fractal generalized Pascal matrices

Zero generalized Pascal matrices include Pascal triangle modulo 2

New system of matrices underpins the fractal structure

Abstract

Set of generalized Pascal matrices whose elements are generalized binomial coefficients is considered as an integral object. The special system of generalized Pascal matrices, based on which we are building fractal generalized Pascal matrices, is introduced. Pascal matrix (Pascal triangle) is the Hadamard product of the fractal generalized Pascal matrices. The concept of zero generalized Pascal matrices, an example of which is the Pascal triangle modulo 2, arise in connection with the system of matrices introduced.