Some Identities on Generalized Arithmetic Triangles

Abstract

Summary We generalize the arithmetic triangle of Blaise Pascal, Yang Hui, and others, by maintaining its recurrence relation, but replacing the traditional 1s on the boundary with arbitrary sequences. Within these structures, we examine some identities commonly studied in Pascal’s version, including those related to sums and alternating sums of entries in rows as well as the so-called hockey stick identities. We use recurrence relations and elementary generating functions to derive general results and see how these results can be used in some interesting special cases.