My motivation to the following question stems from the discussion at Complex Zeroes of Stirling functions of the second kind about the location of the complex zeroes of Stirling functions of the second kind, I am curious in exploring the monotonic increase of a ratio of two consecutive Stirling functions of the second kind.

As a reminder, $S_{(x,n)}=\frac{1}{n!}\sum\limits_{k=1}^{n}{n\choose k}(-1)^{n-k}k^{x}$.

I believe that $f_{n}(x)=\frac{S_{(x,n+1)}}{S_{(x,n)}}$ is an increasing function for increasing real parts of $x$. Furthermore, my motivation for this conjecture stems from the infinite series representation $g(x)=\sum\limits_{n=1}^{\infty}S_{x,n}a_{n}$ with $a_{n}=\sum\limits_{k=1}^{n}s_{(n,k)}g(k)$ where $s_{(n,k)}$ are Stirling numbers of the first kind. There are a class of functions that satisfy this infinite series representation that maps a sequence to a complex valued function.

In the case of $g(x)=\frac{1}{x}$ this infinite series seems to converge for $\Re{(x)}>0$. Furthermore, there is the property of this series that I can shift the series if $g(x)$ converges for $\Re{(x)}>\Re{(c_{1})}$ and write $g(x+c_{2})$ converges for $\Re{(x)}>\Re{(c_{1}-c_{2})}$. There are some series in practice that this infinite series representation works for. It works for the Lerch Transcendent (up to a pole), the Hurwitz Zeta Function (up to a pole), The Riemann Zeta function (up to a pole), a large class of other Dirichlet series (up to a pole), all polynomials, inverse polynomials (up to a pole), exponential functions, and other functions. I am trying to prove that the $\mathcal{S}$-representation of function converges for $\Re{(x)}>\Re{(a)}$ where $a$ is the pole of the function with the largest real part. My conjecture is important because it implies that if an $\mathcal{S}$-representation converges for $x=c_{1}$ it converges for $\Re{(x)}>\Re{(c_{1})}$. There may be a range of exceptions to this conjecture. These exceptions are the exceptions I would like to know about now.

These infinite series I have derived can be shown to be equivalent in some cases to already existing formulas for these functions. For example, in the case of the Riemann Zeta function. By mathematical manipulation it is possible to show that the $S$-representation of $\zeta(2-x)(1-x)$ is equivalent to the slower of Hasses's two globally convergent formulas for the Riemann zeta function that converges everwhere except at the pole $x=1$.

However, it would also be nice even if these representation only work for $x$ real.

Furthermore, I conjecture $f_{n}(x)$ is increasing in value, but decreasing in absolute value for increasing integer $n$. This makes sense because of what it says about when I try to apply the ratio test to the $\mathcal{S}$-representation of $g(x)$.

This is the motivation behind asking this question. Except I have had a lot of trouble with finding a proof of this property of Stirling Functions of the Second Kind. I have tried looking at $\frac{d}{dx}\frac{S_{(x,n+1)}}{S_{(x,n)}}=f_{n}'(x)=\frac{S_{(x,n)}\frac{d}{dx}S_{(x,n+1)}-S_{(x,n+1)}\frac{d}{dx}S_{(x,n)}}{S_{(x,n)}^{2}}=h(x)$.

$h(x)>0$ iff $S_{(x,n)}\frac{d}{dx}S_{(x,n+1)}-S_{(x,n+1)}\frac{d}{dx}S_{(x,n)}>0 \forall x$. Then I can multiply the sum representation for $S_{(x,n)}$, $S_{(x,n+1)}$, and their derivatives. It is easy to see that this function is eventually increasing. However, it is very difficult to show that this function is increasing for all increasing real parts of $x$.

I have also looked at work by Kilbas, Butzer, and Trujillo who define generalized Stirling functions of the second kind in a similar manner for $\Re{(x)}>0$ in order to find another representation for this approximation:

$S_{(x,n)}=\frac{(-1)^{n}}{n!\Gamma{(-x)}}\int\limits_{0}^{\infty}(1-e^{-t})^{n}\frac{dt}{t^{1+x}}$.

Therefore, $f_{n}(x)=\frac{-1}{n+1}\frac{\int\limits_{0}^{\infty}(1-e^{-t})^{n+1}\frac{dt}{t^{1+x}}}{\int\limits_{0}^{\infty}(1-e^{-t})^{n}\frac{dt}{t^{1+x}}}$.

However, it also seems like a difficult task to prove that $f_{n}(x)$ is monotonically increasing for increasing real parts of $x$ in this way. I was wondering if anyone had any feedback or insight?

Thank you in advance for any help. At least in my opinion, this is a difficult question, so I very much appreciate help from anyone who can shed some light on the validity of my conjecture and if it is wrong how wrong it is.

For now the information I am most interested in whether my conjecture holds true for real $x$. This is what has in my experience been most supported by graphical data.

In the real case, I conjecture that $f_{n}(x)$ is a strictly increasing function for all increasing real $x$.