Closed form expression for constants

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!

• this is what i tried : using the definition of the digamma :$$\sum_{m=1}^{\infty}\frac{1}{m(ms-1)}=-\gamma-\psi\left(\frac{s-1}{s}\right)$$. where $\gamma$ is the Euler-Mascheroni constant, we can write Laurent expansions around each $\frac{1}{n}$: $$\digamma\left(\frac{s-1}{s}\right)=-\frac{1}{n(ns-1)}+\sum_{k=0}^{\infty}a_{k,n}\left(s-\frac{1}{n}\right)^{k}$$. making use of the identity: $$\exp\left(\sum_{n=1}^{\infty}a_{n}\frac{(s-s_{0})^{n}}{n!}\right)=\sum_{m=0}^{\infty}\frac{B_{m}\left(a_{1},...,a_{m}\right)}{m!}(s-s_{0})^{m}$$ Commented Feb 11, 2013 at 20:31
• Where $B_{m}\left(a_{1},...,a_{m}\right)$ are the complete Bell polynomials, it becomes fairly easy to compute $c_{k,n}$. However, i couldn't give a formula for the numbers $a_{k,n}$ !! Commented Feb 11, 2013 at 20:36
• sorry, it is : $$\psi\left(\frac{s-1}{s}\right) = -\frac{1}{n(ns-1)}+\sum_{k=0}^{\infty}a_{k,n}\left(s-\frac{1}{n}\right)^{k}$$ Commented Feb 12, 2013 at 21:20
• Don't we always have that $\frac{1}{n(ns-1)}$ term in there regardless of $k$? So evaluation at $s=1/n$ should give us a $0$ denominator. What am I missing? Commented Feb 13, 2013 at 17:43
• yeah, but we are evaluating $\psi\left(\frac{s-1}{s}\right)+\frac{1}{n(ns-1)}$ and its derivatives. the poles of the digamma term at each $\frac{1}{n}$ cancel with those of $\frac{1}{n(ns-1)}$ Commented Feb 13, 2013 at 20:49

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.