From: Gerry Myerson <gerry@mpce.mq.edu.au>
Subject: Re: Values of a power series
Date: Thu, 08 Feb 2001 14:05:03 +1000
Newsgroups: sci.math
Summary: Special (closed-form) values of the dilogarithm function

In article <o90dt4gas3ua@forum.mathforum.com>, Qqqquet@mindspring.com 
(Leroy Quet) wrote:

> Gerry Myerson wrote:
> >....
> >According to Jolley, Summation of Series, p. 66, Series #360, 
> >"x/1^2 + x^2/2^2 + x^3/3^3 + ... can be summed in five cases only: 
> >x = 1, \sum_1^infty x^n/n^2 = (pi^2)/6
> >x = -1, \sum_1^infty x^n/n^2 = (pi^2)/12 
> >x = 1/2, \sum_1^infty x^n/n^2 = (pi^2)/12 - (1/2)(log 2)^2
> >x = 2 sin (pi/10), 
> >    \sum_1^infty x^n/n^2 = (pi^2)/10 - (log (2 sin (pi/10)))^2
> >x = (2 sin (pi/10))^2, 
> >    \sum_1^infty x^n/n^2 = (pi^2)/15 - 2 log (2 sin (pi/10))"
> 
> BTW, 2 sin(pi/10) = (sqrt(5) -1)/2.
> 
> I think I found another one once.
> x = -2 sin(pi/10) = (1-sqrt(5))/2.
> Then the sum is (ln(2 sin(pi/10)))^2/2 - pi^2/15.

New to me but not, I'm afraid, to others. Leonard Lewin is the PR man 
for dilogarithms. In his book, Polylogarithms and Associated Functions, 
North Holland 1981, p.7, he gives the one Quet gives above, and also 
when x = -(1 + sqrt 5)/2, 
the sum is - (pi^2)/10 + (1/2) log^2 ((1 + sqrt 5)/2). [|x| > 1, so 
I guess we're talking about the analytic continuation here] 

Then he says, "These would appear to be the only real values for 
which the Li_2 function can be expressed in terms of more elementary 
ones." 

I take the word "appear" to mean that he has a strong suspicion but 
no proof. 

Lewin returns to the topic in Structural Properties of Polylogarithms, 
Amer. Math. Soc. 1991, p. 2, where he gives what looks to me like the 
same set of evaluations. Well, he gives the evaluations at 1, -1, 1/2, 
rho (where rho = (-1 + sqrt 5)/2), and rho^2, and refers to their 
inversions by the formula 

    Li_2(-z) + Li_2(-1/z) = 2 Li_2(-1) - (1/2) log^2 z, 

and to getting -rho from the formula 

    Li_2(z) + Li_2(-z) = (1/2) Li_2(z^2). 

Again, he says, "These results are the only ones known ... for which 
the dilogarithm can be expressed directly in terms of simpler functions, 
in closed form."

Anyway, Lewin is the place to start if you want to work on dilogarithms 
without reinventing the wheel. 

Gerry Myerson (gerry@mpce.mq.edu.au)