I'm interested in finding sums of the form
where $N$ can be supposed to be constant.
I'm using the difference calculus to help solve these sums efficiently. I believe that I can simulate this sum as
I'm attempting to simulate the sum, since I'm having trouble directly evaluating it any other way. I'm wondering if there are alternatives that may be easier to work with. So my question is basically what's the best way to evaluate or simulate this? Alternatives are welcome, just as any good method is!
This is actually part of a larger formula. In other words, I'm working with something like
So I'm especially interested in something that I can easily work with inside of a larger difference equation.