Solving Generalized Fibonacci Recurrences

We consider finding a Binet formula for the continuous function f:dt—>% property where k is a given integer and k>\\, either f(0),..., f(k-1) are given initial values, which has the /(*) = I/(*-0, (i) l</<Ar or / : [0, k)-> $ is a given continuous initial function where This generalizes the Fibonacci sequence in new ways. Instead of viewing the sequence as an automorphism on the integers, its domain becomes the reals. The Binet formula also allows the initial values to be arbitrary values, possibly complex ones. Instead of having k initial values for the function of order k, we also allow an initial function which is defined on the interval [0, k). When only k initial values are given, there can be many possible functions /. However, the following can be shown by induction. Lemma LI: Given an initial function,/is uniquely defined on 9t, and if then/is continuous. lim/(x) = £ / ( x), x