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