Useful recursion for sum of exponentials?

Question

Given I have a function of form $f(t)=k_1 e^{j_1*t} + k_2 e^{j_2t} +...$, are there any useful recursions of form $f(t+1)=\alpha(f(t))$? I have a lot of accounts with different interest rates to track and I'd like to save memory. I don't think there's anything I can do here, but I want to make sure. If it helps, $0<j_x<12\%$.

Answer 1

Let's say there are $m$ terms: $f(t) = \sum_{i=1}^m k_i r_i^t$ where $r_i = \exp(j_i)$. Then you get a linear recurrence of order $m$: $$ f(t) = \sum_{i=1}^{m} a_i f(t-i)$$ where $$ \prod_{i=1}^m (x - r_i) = x^m - \sum_{i=1}^{m} a_i x^{m-i}$$

EDIT: For example, consider the case $m=2$ with $r_1 = 1$ and $r_2 = 3$. Then $$\prod_{i=1}^m (x - r_i) = (x-1)(x-3) = x^2 - 4 x + 3$$ and the recurrence is

$$f(t) = 4 f(t-1) - 3 f(t-2)$$