Could you explain where this came from: $$\sum _{k=0}^{\infty } (k!)^2 (-y)^k=\frac{G_{1,3}^{3,1}\left(\frac{1}{y}\mid{{0}\atop{0,0,0}}\right)+2 \left(\log \left(\frac{1}{y}\right)+\log (y)\right) K_0\left(\frac{2}{\sqrt{-y}}\right)}{y}$$ where $G$ is Meijer $G$ and $K$ is the modified Bessel function. I tried numerical analysis when $y =1/2$ with Maple and it seems to work.
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Sorry but I don't understand $\sum_{k=0}^{\infty}\left(k!\right)^{2}y^{k}\left(-1\right)^{k}$ doesn't converge... How do you get this equality? – Marco Cantarini Feb 24 '15 at 12:53
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hello Marco a classical divergent Euler Series $$\sum _{k=0}^{\infty } (-1)^k k!=\int_0^{\infty } \frac{e^{-t}}{t+1} , dt$$ – Feb 24 '15 at 19:31
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A related question. – J. M. ain't a mathematician May 01 '15 at 18:12
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@Guesswhoitis., you used to be JM? nice to see you back. – abel May 01 '15 at 18:20