Evaluating a class of infinite sums in closed form
The other day I ran across the surprising identity
and wondered how many sums of this form can be evaluated in closed form like this. Quite a few it turns out.
Sums of the form
evaluate to a rational number when k is a non-negative integer and c is a rational number with |c| > 1. Furthermore, there is an algorithm for finding the value of the sum.
The sums can be evaluated using the polylogarithm function Lis(z) defined as
using the identity
We then need to have a way to evaluate Lis(z). This cannot be done in closed form in general, but it can be done when s is a negative integer as above. To evaluate Li-k(z) we need to know two things. First,
and second,
Now Li0(z) is a rational function of z, namely z/(1 - z). The derivative of a rational function is a rational function, and multiplying a rational function of z by z produces another rational function, so Lis(z) is a rational function of z whenever s is a non-positive integer.
Assuming the results cited above, we can prove the identity
stated at the top of the post.The sum equals Li-3(1/2), and
The result comes from plugging in z= 1/2 and getting out 26.
When k and c are positive integers, the sum
is not necessarily an integer, as it is when k = 3 and c = 2, but it is always rational. It looks like the sum is an integer if c= 2; I verified that the sum is an integer for c = 2 and k = 1 through 10 using the PolyLog function in Mathematica.
Update: Here is a proof that the sum is an integer when n = 2. From a comment by Theophylline on Substack.
The sum is occasionally an integer for larger values of c. For example,
and
