I am looking for a power-series expression of the form $\Gamma(z)=b+\sum_{k=0}^\infty a_kz^k$ where the $a_k$ can be calculated as some function of k.
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4$\begingroup$ $\Gamma$ has a pole at $z = 0$. Do you mean a Laurent series of the form $$\frac{b}{z} + \sum_{k = 0}^{\infty} a_k z^k ?$$ $\endgroup$– Travis WillseJul 7, 2015 at 6:39
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$\begingroup$ It sounds like I DO mean that. $\endgroup$– JacksonFitzsimmonsJul 8, 2015 at 6:13
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1 Answer
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$\Gamma(z)=\dfrac1z+\displaystyle\sum_{k=0}^\infty a_k\cdot z^k,~$ where the terms $a_k$ form this “beautiful” sequence here. :-$)$
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$\begingroup$ Exactly what I was looking for, thank you. $\endgroup$ Jul 8, 2015 at 6:14
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$\begingroup$ Is there a reasonable expression for the general term $a_k$? $\endgroup$ Jul 8, 2015 at 6:49
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$\begingroup$ Also, it's worth noting that $\Gamma$ has an especially nice Laurent series about $z = 1$. $\endgroup$ Jul 8, 2015 at 6:50
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1$\begingroup$ @TravisWillse thank you. I also took Mathematica for a spin: $$\Gamma(s-1)=\frac{1}{s-1}-\gamma +\frac{1}{12} \left(6 \gamma ^2+\pi ^2\right) (s-1)+\frac{1}{6} \left(-\gamma ^3-\frac{\gamma \pi ^2}{2}+\psi ^{(2)}(1)\right)(s-1)^2 +\frac{1}{24} \left(\gamma ^4+\frac{3 \pi ^4}{20}+\gamma ^2 \pi ^2-4 \gamma \psi ^{(2)}(1)\right)(s-1)^3+O\left((s-1)^4\right),$$ where I believe the polygamma's are some function of $\zeta(q)$, $q\in\mathbb{N}$. $\endgroup$– pshmath0Feb 4, 2020 at 8:35