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_{âˆ’k}(*z*) we need to know two things. First,

and second,

Now Li_{0}(*z*) is a rational function of *z*, namely *z*/(1 âˆ’ *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_{âˆ’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