MULTIPLICATION FORMULAS FOR PERIODIC-FUNCTIONS

Carlitz and others have proved that if f is a polynomial such that it satisfies the formula [GRAPHICS] then f is (essentially) the k(th) degree Bernoulli polynomial. The purpose of this paper is to discuss the slightly more general formula [GRAPHICS] when f is periodic with period 1. The notation n(d) under the summation sign indicates that n runs through a complete system of residues mod d . Formulae like (1.1) and (1.2) occur also in theories of Franel's formula and in the elementary theory of Dedekind sums. In this paper we will pretty much characterize the periodic bounded variation solutions of (1.2). Then we provide a weak generalization of the current best form of Franel's theorem, and use this result to provide a method for constructing new solutions of (1.2) from old ones, at least in principle.