Special values of generalized log-sine integrals

Abstract

We study generalized log-sine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of Nielsen polylogarithms at ±1. For general arguments we present algorithmic evaluations involving Nielsen polylogarithms at related arguments. In particular, we consider log-sine integrals at π/3 which evaluate in terms of polylogarithms at the sixth root of unity. An implementation of our results for the computer algebra systems Mathematica and SAGE is provided.

Mahler measures, short walks and log-sine integrals

“…This was one of the motivations for a systematic study of log-sine integrals detailed in [13] and sketched next in Section 4.…”

“…Moreover, the form of the evaluation (13) promises that, for fixed n, the even moments W n (2k) satisfy a linear recurrence with polynomial coefficients. For instance, for n = 3,…”

“…Our emphasis is on automatically provable evaluations. We will only sketch the ideas of such evaluations -details may be found in [13].…”

“…All of these values have been automatically obtained using the routines described and implemented by the authors in [13]. The resulting computer algebra packages can be downloaded from the website of the second author; a link is given in Section 5.…”

“…♦ Searching for a way to generalize the generating function approach used in (72) and (76) to general log-sine integrals Ls n (π) at π, we derived, [13], the generating function…”