Evaluating infinite sums in closed form with polylogarithms

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 Li_s(z) defined as

using the identity

We then need to have a way to evaluate Li_s(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 Li_s(z) is a rational function of z whenever s is a non-negative 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

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