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44
Multiple Polylogarithms: A Brief Survey
"... . 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 devel ..."
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Cited by 28 (8 self)
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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 qanalogue of the shuffle product. 1.
SYMBOLIC SUMMATION ASSISTS COMBINATORICS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 56 (2007), ARTICLE B56B
, 2007
"... We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multisum examples which are related to combinatorial problems. ..."
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Cited by 24 (11 self)
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We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multisum examples which are related to combinatorial problems.
Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5)
, 1998
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Special Values of Multidimensional Polylogarithms
 TRANS. AMER. MATH. SOC
, 1998
"... Historically, the polylogarithm has attracted specialists and nonspecialists 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 highenergy physics. More recently, w ..."
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Cited by 20 (12 self)
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Historically, the polylogarithm has attracted specialists and nonspecialists 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 highenergy 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...
Continued Fractions, Comparison Algorithms, and Fine Structure Constants
, 2000
"... 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 ..."
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Cited by 14 (4 self)
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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.
When is 0.999... equal to 1
 Amer. Math. Monthly
, 2007
"... Abstract. A doubly infinite sum, numerically evaluated at between 0.999 and 1.001, turns out to have a nice value. 1. ..."
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Cited by 9 (5 self)
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Abstract. A doubly infinite sum, numerically evaluated at between 0.999 and 1.001, turns out to have a nice value. 1.
STRUCTURAL THEOREMS FOR SYMBOLIC SUMMATION
"... Starting with Karr’s structural theorem for summation —the discrete version of Liouville’s structural theorem for integration — we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can b ..."
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Cited by 9 (7 self)
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Starting with Karr’s structural theorem for summation —the discrete version of Liouville’s structural theorem for integration — we work out crucial properties of the underlying difference fields. This leads to new and constructive structural theorems for symbolic summation. E.g., these results can be applied for harmonic sums which arise frequently in particle physics.
Some series of the zeta and related functions
 Analysis
, 1998
"... A rather classical (over two centuries old) theorem of Christian Goldbach (16901764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (17001782), was revived in 1986 by Shallit and Zikan [23] as the following problem: ..."
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Cited by 7 (2 self)
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A rather classical (over two centuries old) theorem of Christian Goldbach (16901764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (17001782), was revived in 1986 by Shallit and Zikan [23] as the following problem:
Signed qanalogs of Tornheim’s double series
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 136, 2689–2698, 2008. MARKUS KUBA, INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE, TECHNISCHE UNIVERSITÄT WIEN, WIEDNER HAUPTSTR. 810/104, 1040
, 2008
"... We introduce signed qanalogs of Tornheim’s double series and evaluate them in terms of double qEuler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature. ..."
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Cited by 6 (2 self)
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We introduce signed qanalogs of Tornheim’s double series and evaluate them in terms of double qEuler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature.