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

Abstract: In this paper, we shall first introduce the Pascal determinantal array of order $k$ as a generalization of the standard Pascal array. We also present an algorithm to produce determinantal arrays of any order. Then we investigate geometric properties of these new determinantal arrays. As a by product, we give a proof of the correctness of our proposed algorithm. Then we give a geometric interpretation of the determinant of any $k$ by $k$ subarray of the Pascal array. Finally, we will give a generalization of Rahimpour's determinantal identity, using the above geometric interpretation.