PASCALIAN TRIANGLES AND EXTENSIONS TO HEXAGONS

Abstract

Various properties of binomial coefficients , expressed in terms of patterns in the Fascal Triangle, are generalized in two directions. First, the properties are extended over all integer values values of n and r, adopting a natural definition of in this extended Pascal Hexagon, based either on considerations of symmetry or on interpretations of as coefficients in appropriate (Taylor or Laurent) expansions of (1+x)ⁿ. Second, the properties, suitably adapted, are shown to hold also for the Leibniz, harmonic coefficients and for q-analogues of these coefficients. In all these cases, the extension to the entire hexagon is effected. Certain patterns, giving rise to the Pascal Flower and the (generalized) Star of David Theorem axe shown to occur under very general conditions which include our 4 examples above—and much else.