Closed form expression for constants

Question

We have the constants $c_{k,n}$ defined by : $$c_{k,n}=\frac{d^{k}}{ds^{k}}\left(\frac{e^{\frac{1}{n(ns-1)}}e^{\psi\left(\frac{s-1}{s} \right )}}{s} \right ),$$ where $\psi(s)\;$ is the Digamma function and the derivatives are evaluated at $s=\frac{1}{n}$, $n\in\mathbb{Z}^{+}$. We wish to have a closed-form expression for these constants. I tried Mellin's Formula, but couldn't get answers!

Answer 1

by General Leibniz rule (Higher product rule):

$$\frac{d^k}{ds^k}\left(\frac{e^{\frac{1}{n(ns-1)}}e^{\psi\left(\frac{s-1}{s} \right)}}{s}\right)=$$ $$\sum_{i=0}^k\left(\binom{k}{i}\left(\frac{d^i}{ds^i}e^{\frac{1}{n(ns-1)}}\right)\sum_{j=0}^{k-i}\binom{k-i}{j}\left(\frac{d^j}{ds^j}e^{\psi\left(\frac{s-1)}{s}\right)}\right)(i+j-k)_{-i-j+k}\ s^{i+j-k-1}\right)$$

$(a)_b$: Pochhammer symbol

$$$$

by Faà di Bruno's formula (Higher chain rule):
$\frac{d^i}{ds^i}e^{f(s)}=e^{f(s)}B_i(f(s))$
$B_i()$ is the complete exponential Bell polynomial.
$B_i(f(s))=B_i(f^{(1)}(s),f^{(2)}(s),...,f^{(i)}(s))$

$$\frac{d^i}{ds^i}e^{\frac{1}{n(ns-1)}}=e^{\frac{1}{n(ns-1)}}B_i\left(\frac{1}{n(ns-1)}\right)$$

$$\frac{d^l}{ds^l}\frac{1}{n(ns-1)}=\frac{(-l)_l\ n^l}{n(ns-1)^{l+1}}$$

$$$$

$$\frac{d^j}{ds^j}e^{\psi\left(\frac{s-1)}{s}\right)}=e^{\psi\left(\frac{s-1)}{s}\right)}B_j\left(\psi\left(\frac{s-1)}{s}\right)\right)$$

$$\frac{d^l}{ds^l}\psi\left(\frac{s-1)}{s}\right)=\sum_{t=0}^{2l}(-1)^t\binom{l-1}{t}\binom{l}{t}t!\ \frac{\psi\left(l-t,\frac{s-1}{s}\right)}{s^{2l-t}}$$

$\psi$ is the polygamma function. The coefficients of $\frac{\psi\left(l-t,\frac{s-1}{s}\right)}{s^{2l-t}}$ build the series OEIS: A111884.

Maybe it's possible to derive a formula for the coefficients of the original series on the top.

For $s=\frac{1}{n}$, the series doesn't converge. We have to determine the limit of the series therefore.