In this final post (by the moment) in the polylogia series we will write some additional formulae for polylogs and associated series.
Firstly, we have
and now, if
The next identity also holds
The Bose-Einstein integral can be rewritten for as follows
The polylog has the following asymptotic series. If we have
Euler’s dilogarithm or Spence’s function has nice features. Interestingly, computer algebra systems generally define the dilogarithm as , thus be aware with the definitions you read, use and write. In our case:
Moreover, for you can find
and we have the Abel identity:
Euler’s reflection formula with follows up
The pentagon identity, with and is a mutation of the Abel identity
Landen’s identity is also beautiful. It arises if we write in the Abel identity and we square the relationship we get:
with . We have an inversion formula for this too
with , or
The mathematician Don Zagier has stated the following sentence “(…)The dilogarithm is the only mathematical functionwith a sense of humor(…)”. Have a look at the following values of Euler’s dilogarithm:
Moreover, suppose that is the negative golden mean ratio, then
If we also have
If , we have
There are also polylog ladders much more complex than the above identities. Let us define
Then, a wonderful result by Coxeter (1935) is the next identity
and Landen also derived
Now, we can write some multiplication theorems. The duplication identity
Gauss wrote the next sum, a discrete Fourier transform
The Kummer’s identity for duplication reads
Moreover, if , we can derive the result
and where is the Bessel function with .
Periodic zeta functions are defined by
Finally, we end this series with some Hurwitz zeta function series and identities
See you in my next blog post!
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