Carry propagation in signed digit representations

AbstractVon Neumann’s addition method adds two numbers given in q-ary representation by forming a number consisting of the added digits, reduced modulo q, and another number, representing the carries and repeating this until the string of carries consists only of zeros. The average number of iterations was studied by Knuth.We extend these results by considering the (q,d) system, with base q and digits d,d+1,…,d+q−1, as well as the symmetric signed digit expansions, for even q, with digits −q/2,…,q/2, and a special rule to make representations of integers unique