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Question

Is there any closed formula for the sum $ ~\sum_{k = 1} ^ {n} r^k k^r ~$ where $~r~$ is an integer?

Thank you very much in advance.

I think these are related to the Stirling numbers, but, upon looking them up, the relation is not clear to me. Thought I'd mention it anyway. — marty cohen, Oct 23 '14 at 20:57
The exact formula at https://math.stackexchange.com/questions/4739315/is-there-a-general-formula-for-sum-limits-n-0n-1-qnnm/4739554#4739554? — mocooJ, Jul 21 '23 at 01:56

Anatoly · Accepted Answer · 2014-10-23T20:21:40.037

2

The sum corresponds to the following polylogarithm:

$$ \sum_{k = 1} ^ {n} r^k k^r = \operatorname{Li}_{-r}(r)$$

edited Oct 23 '14 at 20:21

answered Oct 23 '14 at 20:17

Anatoly

But polylogarithm is a sum of infinite series, isn't it? – Igor Oct 23 '14 at 21:18
Yes. There are no closed forms in the strict sense for the general polylogarithm. – Anatoly Oct 23 '14 at 21:37
So there' little hope to find exact formula even for this specific sum? – Igor Oct 23 '14 at 21:49
No undefined power sum have an exact formula @Igor, see Falhauber formula. The best approach maybe a desired level of precision on computing, something like the h function. – Masacroso Oct 23 '14 at 22:51
Unfortunately, exact formulas exist only for some specific polylogarithms. For pratical purposes, you could consider the use of fast computation algorithms approximating $\operatorname{Li_n}(z)$ (see for example http://www.mathworks.com/matlabcentral/fileexchange/37229-enhanced-computation-of-polylogarithm-aka-de-jonquieres-function). – Anatoly Oct 24 '14 at 07:06

1 Answers1