A Curious Identity on Multiple Sums over Fields with Applications

Abstract

Let [Formula: see text] be a field, and let [Formula: see text] be integers such that [Formula: see text] and [Formula: see text]. We show that for any subset [Formula: see text], the curious identity [Formula: see text] holds with [Formula: see text] being the set of nonnegative integers. As an application, we prove that for any subset [Formula: see text] with [Formula: see text] being the finite field of [Formula: see text] elements and [Formula: see text] being integers such that [Formula: see text] and [Formula: see text], [Formula: see text] Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996, we show that if [Formula: see text] is an integer with [Formula: see text], then for any subset [Formula: see text] we have [Formula: see text] This implies [Formula: see text].