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. We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral representation for the general multiple polylogarithm, and develop a q-analogue of the shuffle product. 1.

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including a longstanding conjec...

We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multi-sum examples which are related to combinatorial problems.

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There are known algorithms based on continued fractions for comparing fractions and for determining the sign of 2x2 determinants. The analysis of such extremely simple algorithms leads to an incursion into a surprising variety of domains. We take the reader through a light tour of dynamical systems (symbolic dynamics), number theory (continued fractions), special functions (multiple zeta values), functional analysis (transfer operators), numerical analysis (series acceleration), and complex analysis (the Riemann hypothesis). These domains all eventually contribute to a detailed characterization of the complexity of comparison and sorting algorithms, either on average or in probability.

The three dots in the title do not refer to an infinite sequence of 9’s, but to digits that are increasingly hard to compute. The question is philosophical: how many 9’s do we need to see before we start to believe that the constant we are computing

A rather classical (over two centuries old) theorem of Christian Goldbach (1690-1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700-1782), was revived in 1986 by Shallit and Zikan [23] as the following problem:

We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra package Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations. 1.

ABSTRACT. Bailey and Crandall [4] recently formulated “Hypothesis A”, a general principle to explain the (conjectured) normality of the binary expansion of constants like π and other related numbers, or more generally the base b expansion of such constants for an integer b ≥ 2. This paper shows that a basic mechanism underlying their principle, which is a relation between single orbits of two discrete dynamical systems, holds for a very general class of representations of numbers. This general class includes numbers for which the conclusion of “Hypothesis A” is not true. The paper also relates the particular class of arithmetical constants treated by Bailey and Crandall to special values of G-functions, and points out an analogy of “Hypothesis A ” with Furstenberg’s conjecture on invariant measures. AMS Subject Classification(2000): 11K16 (Primary) 11A63, 28D05, 37E05 (Secondary) Keywords: dynamical systems, invariant measures, G-functions, polylogarithms, radix expansions