Lerch's Phi transcendent is
$$ \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} $$
I am trying to compute this for the following parameters:
- $z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (typical value for $|z|$ would be $0.99$ to $0.999$)
- $s$ is real and positive, and $0 \leq s \leq 10$ or so, with the most important region being something like $0 \leq s \leq 2$
- $a$ is real and positive, and $1 \leq a \leq 3000$ or so, although the most important value is $a=1$.
At least for these parameters, after trying a few simple methods, the basic Aitken/Shanks delta-squared acceleration seems to be the fastest. So far I've compared Aitken delta-squared, Wynn's epsilon, and Levin's u-transformation.
There are a few methods mentioned in the literature, most notably the "Combined Nonlinear Condensation Transformation" which is a Van Wijngaarden transformation followed by a Levin u-transformation. However, this only seems to be for real $z$. Some example papers can be seen here and here.
Question: as of 2019, what is the current fastest method to sum this series?
FWIW, this (perhaps unusual) set of parameters has arisen naturally in the setting of audio synthesis, where it is useful for generating anti-aliased digital waveforms such as sawtooth waves, parabolic waves, etc. For instance, if we look at
$$ f(z,s,h) = z\Phi(z, s, 1) - z^{h+1}\Phi(z, s, h+1) $$
then we have $$ f(z,s,h) = \sum_{k=1}^h \frac{z^k}{k^s} $$
so we get something like a "partial polylogarithm," expressible as the difference of two Lerch's transcendents, each of which can be accelerated with something like Aitken's delta-squared.
As a result, we have $\Re[(-i)^s f(e^{2\pi f i t},s, h)]$ is a waveform with exactly $h$ harmonics, with the $k$'th harmonic at amplitude $1/k^s$. So if we set $s=1$ we get the first $h$ harmonics of a sawtooth wave, if we set $s=2$ we get the first $h$ harmonics of a parabolic wave, etc. This is basically just Hurwitz's theorem, but where we then subtract this second Lerch's transcendent term so as to cause all the harmonics beyond $h$ to cancel.
That first term in $f(z,s,h)$ is also equal to the polylogarithm, so perhaps if anyone has any good insight into that, it might be possible to relate to Lerch's transcendent. The end result is also expressible as a Hurwitz's zeta term minus the Lerch term.