A systematic approach to matrix forms of the Pascal triangle: The twelve triangular matrix forms and relations

This work initiates a systematic investigation into the matrix forms of the Pascal triangle as mathematical objects in their own right. The present paper is especially devoted to the so-called G-matrices, i.e. the set of the twelve (n+1)×(n+1)(n+1)×(n+1) triangular matrix forms that can be derived from the Pascal triangle expanded to the level n(2≤n∈N). For n=1n=1, the G-matrix set reduces to a set of four distinct matrices. The twelve G-matrices are defined and the classic Pascal recursion is reformulated for each of the twelve G-matrices. Three sets of matrix transformations are then introduced to highlight different relations between the twelve G-matrices and for generating them from appropriately chosen subsets.