Pascal Determinantal Arrays and a Generalization of Rahimpour's Determinantal Identity

Abstract

We introduce a new infinite family of arrays, the \emph{Pascal determinantal arrays} of order $k$, denoted $PD_k$, which generalize the classical Pascal array via determinantal constructions. We present a recursive algorithm for generating $PD_k$, establish its correctness using Dodgson's condensation and a weighted sliding-cross rule, and provide a geometric interpretation of the entries $P^{(k)}_{i,j}$ as weighted double sticks in the Pascal plane. Our main result proves a conjecture that generalizes Rahimpour's identity: for all $i,j,k \ge 0$, \[ P^{(k)}_{i,j} = P^{(j)}_{i,k}, \] where $P^{(k)}_{i,j}$ is the determinant of the $k \times k$ subarray of the Pascal array starting at $(i,j)$. The proof combines algebraic recurrence techniques with a visual, geometry-based argument that reveals the intrinsic symmetry of determinantal Pascal structures.