205 research outputs found
Parametric Euler Sum Identities
We consider some parametrized classes of multiple sums first studied by
Euler. Identities between meromorphic functions of one or more variables
generate reduction formulae for these sums.Comment: 12 page
Log-sine evaluations of Mahler measures, II
We continue the analysis of higher and multiple Mahler measures using
log-sine integrals as started in "Log-sine evaluations of Mahler measures" and
"Special values of generalized log-sine integrals" by two of the authors. This
motivates a detailed study of various multiple polylogarithms and worked
examples are given. Our techniques enable the reduction of several multiple
Mahler measures, and supply an easy proof of two conjectures by Boyd.Comment: 35 page
Asymptotic relationships between Dirichlet series
AbstractIt is shown, inter alia, that under certain conditions the asymptotic relationhip Σn=1∞ansse−λnx∼lΣn=1∞ane−λnx as x → 0+ between two Dirichlet series implies the same relationship with λn replaced by λnc, 0 < C < 1
Special values of multiple polylogarithms
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 an intriguing conjecture of Don Zagier
Duality for Finite Multiple Harmonic q-Series
We define two finite q-analogs of certain multiple harmonic series with an
arbitrary number of free parameters, and prove identities for these q-analogs,
expressing them in terms of multiply nested sums involving the Gaussian
binomial coefficients. Special cases of these identities--for example, with all
parameters equal to 1--have occurred in the literature. The special case with
only one parameter reduces to an identity for the divisor generating function,
which has received some attention in connection with problems in sorting
theory. The general case can be viewed as a duality result, reminiscent of the
duality relation for the ordinary multiple zeta function.Comment: 12 pages AMSLaTeX. Submitted for publication October 26, 2003;
revised September 14, 2004. New title reflects change in emphasis and new
section devoted to connections with inverse pairs and Hoffman duality.
References added and typos correcte
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
Experimental determination of Apery-like identities for zeta(2n+2)
We document the discovery of two generating functions for the Riemann zeta
values zeta(2n+2), analogous to earlier work for zeta(2n+1) and zeta(4n+3).
This continues work initiated by Koecher and pursued further by Borwein,
Bradley and others.Comment: 15 pages, AMSLaTeX, Proof of Theorem 1 improve
Central Binomial Sums, Multiple Clausen Values and Zeta Values
We find and prove relationships between Riemann zeta values and central
binomial sums. We also investigate alternating binomial sums (also called
Ap\'ery sums). The study of non-alternating sums leads to an investigation of
different types of sums which we call multiple Clausen values. The study of
alternating sums leads to a tower of experimental results involving
polylogarithms in the golden ratio. In the non-alternating case, there is a
strong connection to polylogarithms of the sixth root of unity, encountered in
the 3-loop Feynman diagrams of {\tt hep-th/9803091} and subsequently in
hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.Comment: 17 pages, LaTeX, with use of amsmath and amssymb packages, to appear
in Journal of Experimental Mathematic
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