What is the Closed Form of $\sum_{n = a}^b k^{f(n)}$?

Asked Jul 21 '22 at 18:04

Active Jul 21 '22 at 18:04

Viewed 50 times

I recently used discrete calculus to derive the following formula: $$ \sum_{n = a}^b k^n = \frac{k^{b + 1} - k^a}{k - 1}. $$

I know that the simple chain rule does not work for discrete calculus (and so there is no u-substituion analogue that I am aware of), so I am unable to derive a closed-form formula for the following expression: $$ \sum_{n = a}^b k^{f(n)}. $$

My question is whether or not the above expression has a closed form and what that closed form is. I know that the expression may not have a closed form in general, but I am wondering if there are certain types of functions and/or sequences that allow the above sum to have a closed form.

asked Jul 21 '22 at 18:04

William Ryman

1

Well, if $f$ is a linear function, you can do the same trick. If it is quadratic, perhaps looking at theta functions might tell you some stuff.. – Phicar Jul 21 '22 at 18:35
2

Make $f(n)=\log_k(g(n))$ to ask which $\sum\limits_{n=a}^b g(n)$ have a closed form? Try searching summatory function – Тyma Gaidash Jul 21 '22 at 19:23

What is the Closed Form of $\sum_{n = a}^b k^{f(n)}$?

0 Answers0