Generalized Pascal functional matrix and its applications

In this paper, we introduce the generalized Pascal functional matrix and show that the existing variations of Pascal matrices are special cases of this generalization. We study some algebraic properties of such generalized Pascal functional matrices. In addition, we demonstrate a direct application of these properties by deriving several novel combinatorial identities and a nontraditional approach for LU decompositions of some well-known matrices (such as symmetric Pascal matrices).