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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!

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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: $$\exp\left(\sum_{n=1}^{\infty}a_{n}\frac{(s-s_{0})^{n}}{n!}\right)=\sum_{m=0}^{‌​\infty}\frac{B_{m}\left(a_{1},...,a_{m}\right)}{m!}(s-s_{0})^{m}$$ –  Mohammad Al Jamal Feb 11 '13 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 $a_{k,n}$ !! –  Mohammad Al Jamal Feb 11 '13 at 20:36
    
sorry, it is : $$\psi\left(\frac{s-1}{s}\right) = -\frac{1}{n(ns-1)}+\sum_{k=0}^{\infty}a_{k,n}\left(s-\frac{1}{n}\right)^{k}$$ –  Mohammad Al Jamal Feb 12 '13 at 21:20
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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? –  Alexander Gruber Feb 13 '13 at 17:43
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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 $\frac{1}{n(ns-1)}$ –  Mohammad Al Jamal Feb 13 '13 at 20:49

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