# LOG#162. Polylogia flashes(IV).

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

(1)

and now, if

(2)

(3)

The next identity also holds

The Bose-Einstein integral can be rewritten for as follows

and

The polylog has the following asymptotic series. If we have

(4)

(5)

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:

(6)

Moreover, for you can find

(7)

(8)

and we have the Abel identity:

(9)

Euler’s reflection formula with follows up

(10)

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

with .

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

and

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

(11)

and Landen also derived

(12)

Now, we can write some multiplication theorems. The duplication identity

(13)

Gauss wrote the next sum, a discrete Fourier transform

(14)

The Kummer’s identity for duplication reads

(15)

Moreover, if , we can derive the result

and where is the Bessel function with .

Periodic zeta functions are defined by

(16)

and thus

Finally, we end this series with some Hurwitz zeta function series and identities

See you in my next blog post!

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