Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions

Given a recurrence sequence H, with $$H_n = c_1 H_{n-1} + \dots + c_t H_{n-t}$$ where $$c_i \in \mathbb {N}_0$$ for all i and $$c_1, c_t \ge 1$$ , the generalized Zeckendorf decomposition (gzd) of $$m \in \mathbb {N}_0$$ is the unique representation of m using H composed of blocks lexicographically less than $$\sigma = (c_1, \dots , c_t)$$ . We prove that the gzd of m uses the fewest number of summands among all representations of m using H, for all m, if and only if $$\sigma $$ is weakly decreasing. We develop an algorithm for moving from any representation of m to the gzd, the analysis of which proves that $$\sigma $$ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form $$v_0 H_n + \dots + v_\ell H_{n-\ell }$$ converge in a suitable sense as $$n \rightarrow \infty $$ ; furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if $$\sigma $$ is not weakly decreasing.