TITLE: $n$th derivative of $e^{1/x}$
QUESTION [64 upvotes]: I am trying to find the $n$'th derivative of $f(x)=e^{1/x}$. When looking at the first few derivatives I noticed a pattern and eventually found the following formula
$$\frac{\mathrm d^n}{\mathrm dx^n}f(x)=(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 n+k}$$
I tested it for the first $20$ derivatives and it got them all. Mathematica says that it is some hypergeometric distribution but I don't want to use that. Now I am trying to verify it by induction but my algebra is not good enough to do the induction step.
Here is what I tried for the induction (incomplete, maybe incorrect)
$\begin{align*}
\frac{\mathrm d^{n+1}}{\mathrm dx^{n+1}}f(x)&=\frac{\mathrm d}{\mathrm dx}(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 n+k}\\
&=(-1)^n e^{1/x} \cdot \left(\sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} (-2n+k) x^{-2 n+k-1}\right)-e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k} x^{-2 (n+1)+k}\\
&=(-1)^n e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k}((-2n+k) x^{-2 n+k-1}-x^{-2 (n+1)+k)})\\
&=(-1)^{n+1} e^{1/x} \cdot \sum _{k=0}^{n-1} k! \binom{n}{k} \binom{n-1}{k}(2n x-k x+1) x^{-2 (n+1)+k}
\end{align*}$
I don't know how to get on from here.

REPLY [2 votes]: For $i\in\mathbb{N}$ and $t\ne0$, we have
\begin{equation}\label{exp-frac1x-expans}\tag{1}
\bigl(\textrm{e}^{1/t}\bigr)^{(i)}=\textrm{e}^{1/t}\frac{(-1)^i}{t^{2i}}\sum_{k=0}^{i-1}\binom{i}{k}\binom{i-1}{k}{k!}t^{k}.
\end{equation}
To the best of my knowledge, an inductive proof of the derivative formula \eqref{exp-frac1x-expans} was published on pages 123--124, Section 2.1, Theorem 2.1 of the paper [1] below. Hereafter, there have been a number of literature, for example, the papers [2, 3] below and those collected at a site on wordpress, dedicated to the investigation of the function $\textrm{e}^{1/t}$, the derivative formula \eqref{exp-frac1x-expans}, and their applications to several areas in mathematics.
References

Xiao-Jing Zhang, Feng Qi, and Wen-Hui Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, International Journal of Open Problems in Computer Science and Mathematics 5 (2012), no. 3, 122--127; available online at https://doi.org/10.12816/0006128.
Siad Daboul, Jan Mangaldan, Michael Z. Spivey, and Peter J. Taylor, The Lah numbers and the $n$th derivative of $e^{1/x}$, Mathematics Magazine 86 (2013), no. 1, 39–47; available online at https://doi.org/10.4169/math.mag.86.1.039.
Khristo N. Boyadzhiev, Lah numbers, Laguerre polynomials of order negative one, and the $n$th derivative of $\exp(1/x)$, Acta Universitatis Sapientiae Mathematica 8 (2016), no. 1, 22–31; available online at https://doi.org/10.1515/ausm-2016-0002.