I'm trying to find the asymptotic behavior of the second derivative of the Laguerre polynomial (more precisely, the associated analytic function), $\frac{\partial}{\partial n^2}L_{n}(z)$, as $z\to -\infty$, at $n = -1$. An explicit formula valid for at least $z<0$ would be great, but it's the asymptotic behavior I particularly care about.
I've managed to reduce it to
$$\frac{\partial}{\partial n^2}L_{n}(z)\biggr|_{n=-1} = \sum_{k=1}^{\infty} \biggl[-\frac{\pi^2}{6} + \psi'(k + 1) + H_k^2\biggr] \frac{z^k}{k!}$$
where $\psi'(z) = \frac{\partial^2}{\partial z^2}\ln\Gamma(z)$ is the polygamma function and $H_k = \sum_{i=1}^{k} \frac{1}{k}$ is a harmonic number.
Clearly summing the first term of the three gives $-\frac{\pi^2}{6}e^z$. I think this is valid, given that the sum of that term by itself is convergent and finite for all finite $z$. (right?)
Are there known formulas for the other two terms on the right? Any way to simplify them that doesn't involve an infinite sum, or at least can be queried for its asymptotic behavior as $z\to -\infty$? I've found formulas for $\sum H_k^2/k^2$ and $\sum H_k^2/(k+1)^2$, as well as $\sum H_k z^k/k!$ ($H_k$ not squared), but I can't figure out how to apply any of those to my problem.
If someone has a totally different way to see the large-$z$ behavior of the function, I'd be interested in that too.
Well, there exists an asymptotic series expansion for the Laguerre function
$$ \ell_n(z)~:=~e^{-z} L_n(z)$$ $$\tag{1}~\sim~ \frac{z^{-n-1}}{\Gamma(-n)} {}_2 F_{0}(n+1,n+1;\frac{1}{z})+\frac{(-z)^n e^{-z}}{\Gamma(n+1)} {}_2 F_{0}(-n,-n;-\frac{1}{z}) $$
for $|z|\to\infty$.
The second term in eq. (1) vanishes because the $\Gamma$ function has a pole at $0$, even if we differentiate a finite number of time wrt. $n$. So the second terms in eq. (1) does not contribute to the sought-for limit.
For the leading behaviour, we can replace the generalized hypergeometric function ${}_2 F_{0}$ with $1$. Two differentiations wrt. the real variable $n$ leads to
$$\tag{2} \left. \frac{\partial^2 \ell_n(z)}{\partial n^2} \right|_{n=-1} ~\sim~ \gamma^2 -\frac{\pi^2}{6}+2 \gamma \log(z) +\log^2(z) +O(\frac{1}{z})\quad\text{for} \quad|z|\to\infty.$$