From: Gerry Myerson 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 , 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)