Series power function over exponential function

Question

A typical exercise from calculus is to show that any exponential function eventually grows faster than any power function, i.e. $$ \lim_{k \to \infty} \frac{k^a}{b^k} = 0 \qquad \text{ for } a,b>1.$$ In fact, by the ratio test, we can show for $x=a=b$ the even stronger result that the series $$ \sum_{k=1}^\infty \frac{k^x}{x^k} $$ converges for any $x \in (1,\infty)$. This gave me the idea to consider the function $F\colon (1,\infty) \to \mathbb{R}$ defined by $$ F(x) = \sum_{k=1}^\infty \frac{k^x}{x^k} \qquad \text{for } x \in (1,\infty).$$ Now I am curious what properties I can find for this function, but my literature search so far didn't give really fitting results. Is there a name for this function?

For integer arguments I could already use the relation $$ F(n) = \text{Li}_{-n}\left(\frac{1}{n}\right), \qquad n \in \mathbb{N}$$ with $\text{Li}$ the polylogarithm to find the representation $$ F(n) = \frac{n}{(n-1)^{n+1}}A_n(n), $$ where $A_n$ is the $n$-th Eulerian polynomial. Furthermore, $F$ seems to have a global minimum at around $$ x = 3.1200906359597\ldots \quad \text{with} \quad F(x)=4.1125402415512\ldots$$ that I found by bisection. The above results gave me hope that there is a closed formula for this minimum as well, e.g. something in terms of elementary functions, but I can't really figure it out. Any ideas?

Answer 1

I think this is a special case of the Lerch transcendent, defined as $$ \Phi(z,s,\alpha)=\sum_{n=0}^\infty\frac{z^n}{(n+\alpha)^s}. $$ Specifically, your proposed function $F$ is given by $$ F(x)=\Phi\big(\tfrac{1}{x},-x,0\big). $$ Plotting this function on WolframAlpha confirms that the global minimum you computed is (approximately) correct (unless you want to extend the domain and muck around with complex numbers).

These slides also discuss the so-called fractional polylogarithm $$ \zeta(s,x)=\Phi(x,s,0)=\sum_{n=0}^\infty\frac{x^n}{n^s}. $$ (I think the slides have a typo—sums should start at $n=1$.) The slides indicate that this function has been studied since at least the late nineteenth century. In terms of the fractional polylogarithm, your function is given by $$ F(x)=\zeta\big(-x,\tfrac{1}{x}\big). $$ I don’t know much about these kinds of functions, but it seems like there’s plenty of literature that may answer some of your questions.