Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k$ at $\small x=-1$) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T15:17:33Z http://mathoverflow.net/feeds/question/87113 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87113/alternating-sums-of-powers-of-the-lngamma-small-f-px-sum-logkp-xk Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k$ at $\small x=-1$) Gottfried Helms 2012-01-31T07:23:43Z 2012-01-31T11:34:03Z <p>I'm still fiddling with <a href="http://mathoverflow.net/questions/84958/85010#85010" rel="nofollow">this recent question</a> and come to a detail, whether I can find closed forms for the sums of the $\small lngamma()$ function. Precisely<br> $$\small f_p(x) = \sum_{k=0}^{\infty} \log(k!)^p \cdot x^k$$ and my first question is to find closed forms for $\small f_p(-1)$ for consecutive <em>p</em>. </p> <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 <em>p=0</em> this is $\small \eta(0)$ (the "Dirichlet's eta", or "alternating zeta", function), for <em>p=1</em> 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. </p> <p>The numerical values for the first few <em>p</em> seem to be (using Pari/GP sumalt-procedure)<br> $\small \qquad \begin{array} {rl} p &amp; f_p(-1) \\ \hline \\ 0 &amp; 0.500000000000 \\ 1 &amp; 0.112895676322 \\ 2 &amp; -0.0380319653072 \\ 3 &amp; 0.0135052530749 \\ 4 &amp; 0.0183298626301 \\ 5 &amp; -0.107164190642 \\ 6 &amp; 0.331363715855 \\ 7 &amp; -0.482387386451 \\ 8 &amp; -2.91602127867 \\ 9 &amp; 32.5904726686 \\ 10 &amp; -154.360744999 \\ 11 &amp; -162.033212532\\ \end{array}$ </p>