Laurent series of the function $M(s) = \dfrac{(1-s)[\Gamma(s)\Gamma(1-s)]^2}{(2+s)(1+2s)(3+2s)} $

Question

The function has double poles at $ s = 2,3,\ ... $ (and at other points as well but I am interested only at these points.) Its given that the principal part of Laurent series at these points $ s = n, \ n \in N, \ n \geq 2 $ is

$$ \left[\dfrac{1}{(s-n)}\dfrac{-45 + 28(n-1)^2 + 8(n-1)^3}{[(2+n)(1+2n)(3+2n)]^2)} + \dfrac{1}{(s-n)^2} \dfrac{-(n-1)}{(2+n)(1+2n)(3+2n)} \right] $$

I would like to know how to obtain this.

Reference: Asymptotics of Feynman diagrams and the Mellin-Barnes representation, Physics Letters B 628 (2005) 73-84

Answer 1

Substituting Euler's reflection formula $$ \Gamma(s)\Gamma(1-s) = \frac{\pi}{\sin{(\pi s)}} $$ it is a straightforward calculation using the limit formula for poles' coeffitients.