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Does anyone know of a closed formula for the function $f_k(x)=\sum_{n=1}^{\infty}{n^k x^n}$ ? That is, the generating function of the sequence $1^k,2^k,3^k...$.

It is not hard to see that $f_k(x)=\frac{P(x)}{(1-x)^{k+1}}$, where $P(x)$ is a monic polynomial of degree $k$ (this follows from the identity $f_k(x)=x\cdot f'_{k-1}(x)$ ). A closed formula for the coefficients of $P(x)$ would be very helpful.

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up vote 11 down vote accepted

Your polynomials are $x$ times the Eulerian polynomials.

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Yes, this is the polylog function $\text{Li}_{-k}(x).$

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Thank you very much! – user85783 Jan 26 at 20:37

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