Note on the Greatest Integer Function

In this note we wish to record certain finite sums involving the greatest integer function E(x), which seem to be of some interest. Hermite* has shown that the generating function for E(x) has the simple form, (1) xb ÷(1 -x)(1 -xa) = Σ(n)E[(n + a - b) ÷ a]xn, where a, b are positive integers. To him is due, likewise, the development (2) ) xb ÷(1 -x)(1 + xa) =Σ(n)E1[(n + a - b) ÷ 2a]xn, where (3) E1(x) = E(2x - 2E(x) = E(x + ½) - E(x). As indicated by Hermite, these developments used in conjunction with the expansions for certain theta quotients, yield results of interest in the theory of numbers. Of particular importance are his results expressing some of Kronecker\u27s class number sums (which arise from certain theta constants of the third degree) in terms of E(x)