Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

2012-01-31T07:23:43Z

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely
$$ \small f_p(x) = \sum_{k=0}^{\infty} \log(k!)^p \cdot x^k $$ and my first question is to find closed forms for $\small f_p(-1) $ for consecutive p.

They are all non-converging series, but which can nicely be summed by, for instance, Euler-summation, but I wish to find closed forms (or simple forms of other series with more known analytical properties). For p=0 this is $\small \eta(0) $ (the "Dirichlet's eta", or "alternating zeta", function), for p=1 this is $\small \eta(0)' $ and I expected, that $\small f_2(-1) $ would be something composed by the square of $\small f_1(-1)$ or the second derivative of the $\small \eta $ at zero, but didn't succeed so far.

The numerical values for the first few p seem to be (using Pari/GP sumalt-procedure)
$\small \qquad \begin{array} {rl} p & f_p(-1) \\ \hline \\ 0 & 0.500000000000 \\ 1 & 0.112895676322 \\ 2 & -0.0380319653072 \\ 3 & 0.0135052530749 \\ 4 & 0.0183298626301 \\ 5 & -0.107164190642 \\ 6 & 0.331363715855 \\ 7 & -0.482387386451 \\ 8 & -2.91602127867 \\ 9 & 32.5904726686 \\ 10 & -154.360744999 \\ 11 & -162.033212532\\ \end{array} $

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$) - MathOverflow

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)