Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$? It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential function $x^x$ exist somewhere?
What is the general formula for the $n$th derivative of the power-exponential function $x^x$?
 A: The Stirling numbers of the first kind $s(n,k)$ for $n\ge k\ge0$ can be analytically generated by
\begin{equation*}%\label{1st-stirl-gen-funct}
\frac{[\ln(1+t)]^k}{k!}=\sum_{n=k}^\infty s(n,k)\frac{t^n}{n!},\quad |t|<1,
\end{equation*}
which can be rearranged as Maclaurin's power series expansion of the power function
\begin{equation}\label{1st-stirl-gen-funct-rewr}
\biggl[\frac{\ln(1+t)}{t}\biggr]^k=\sum_{n=0}^\infty \frac{s(n+k,k)}{\binom{n+k}{k}}\frac{t^{n}}{n!}, \quad |t|<1, \quad k\ge0.
\end{equation}
Comtet's numbers $b(n,k)$ are generated by
\begin{equation}\label{b(n-k)-gen-eq}
\frac{[(1+t)\ln(1+t)]^k}{k!}=\sum_{n=k}^{\infty}b(n,k)\frac{t^n}{n!},\quad k\ge0.
\end{equation}
For $n\in\mathbb{N}_0=\{0,1,2,\dotsc\}$, we have
\begin{equation}\label{power-exp-deriv-eq}
(x^x)^{(n)}=n!x^{x-n}\sum_{k=0}^{n} x^{k} \sum_{j=0}^{k}\Biggl[\sum_{q=0}^{n-k} \frac{s(q+j,j)}{(q+j)!} \binom{j}{n-k-q}\Biggr]\frac{(\ln x)^{k-j}}{(k-j)!}
\end{equation}
and
\begin{equation}\label{power-exp-deriv-b(n-k)}
(x^x)^{(n)}
=x^{x-n}\sum_{k=0}^{n} x^{k} \sum_{j=0}^{k}(-1)^{j}\binom{j-n-1}{j} b(n-j,k-j)(\ln x)^{j}.
\end{equation}
Consequently, Taylor's power series expansion around $x=1$ is
\begin{equation}\label{power-exp-taylor-ser}
x^x=\sum_{n=0}^\infty\Biggl[\sum_{k=0}^{n} \sum_{q=k}^{n} \frac{s(q,k)}{q!} \binom{k}{n-q}\Biggr](x-1)^n, \quad |x-1|<1
\end{equation}
and
\begin{equation}\label{b(n-k)-ser=1-eq}
x^x=\sum_{n=0}^\infty\Biggl[\sum_{k=0}^{n} b(n,k)\Biggr]\frac{(x-1)^n}{n!}, \quad |x-1|<1.
\end{equation}
These texts are extracted from Theorem 1 of the following paper:

*

*Jian Cao, Feng Qi, and Wei-Shih Du, Closed-form formulas for the $n$th derivative of the power-exponential function $x^x$, Symmetry 15 (2023), no. 2, Article 323, 13 pages; available online at https://doi.org/10.3390/sym15020323.

