Hypergeometric Series and Multiple Polylogarithms

Abstract. Identities for multiply nested series extending the multiple zeta values introduced by Zagier will be discussed. A new "Clausen- like" factorization result for Gaussian hypergeometric series is established and extended to solutions of a broader class of second order differential equations. This leads to the resolution of several previously conjectured evaluations for certain multiple polylogarithms with arbitrarily many arguments. An intriguing conjecture concerning evaluations for a parametrized family of alternating multiple polylogarithmic values remains unproved. I will outline the progress to date on this conjecture. After building up sufficient theory, it is possible to give "one-line" proofs of the first instance. The technique suggests that increasingly complicated "one-line" proofs for any given instance should be possible.