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Experimental Mathematics: Recent Developments and Future Outlook
 CECM PREPRINT 99:143] FFL J.M. BORWEIN AND P.B. BORWEIN, "CHALLENGES FOR MATHEMATICAL COMPUTING," COMPUTING IN SCIENCE & ENGINEERING, 2001. [CECM PREPRINT 01:160
, 2000
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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 19 (6 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.
Special values of generalized logsine integrals
"... We study generalized logsine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of multiple polylogarithms at ±1. For general arguments we present algorithmic evaluations involving polylogarithms at related arguments. In particular, we consider logsi ..."
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Cited by 6 (5 self)
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We study generalized logsine integrals at special values. At π and multiples thereof explicit evaluations are obtained in terms of multiple polylogarithms at ±1. For general arguments we present algorithmic evaluations involving polylogarithms at related arguments. In particular, we consider logsine 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. 1.
Experimental Determination of ApéryLike Identities for ζ(2n + 2)
, 2006
"... We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1 ..."
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Cited by 5 (0 self)
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We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1
A note on polyBernoulli numbers and multiple zeta values
"... Abstract. We review several occurrences of polyBernoulli numbers in various contexts, and discuss in particular some aspects of relations of polyBernoulli numbers and special values of certain zeta functions, notably multiple zeta values. ..."
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Cited by 1 (0 self)
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Abstract. We review several occurrences of polyBernoulli numbers in various contexts, and discuss in particular some aspects of relations of polyBernoulli numbers and special values of certain zeta functions, notably multiple zeta values.
Mahler measures, short walks and logsine integrals
, 2012
"... The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years — very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of logsine integrals w ..."
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Cited by 1 (1 self)
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The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years — very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of logsine integrals we provide systematic evaluations of various higher and multiple Mahler measures. The evaluations in terms of logsine integrals become particularly useful in light of the fact that logsine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for logsine integrals. Our initial interest in considering Mahler measures stems from a study of uniform random walks in the plane as first introduced by Pearson. The main results on the moments of the distance traveled by an nstep walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures.
Logsine Evaluations of Mahler measures This paper is dedicated to the memory of Alf van der Poorten
, 2010
"... We provide evaluations of several recently studied higher and multiple Mahler measures using logsine integrals. We then give additional related evaluations that are made possible by our methods. We also explore related generating functions for the logsine integrals. 1 ..."
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We provide evaluations of several recently studied higher and multiple Mahler measures using logsine integrals. We then give additional related evaluations that are made possible by our methods. We also explore related generating functions for the logsine integrals. 1
Integrals and Polygamma Representations for Binomial Sums
"... We consider sums involving the product of reciprocal binomial coefficient and polynomial terms and develop some double integral identities. In particular cases it is possible to express the sums in closed form, give some general results, recover some known results in Coffey and produce new identitie ..."
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We consider sums involving the product of reciprocal binomial coefficient and polynomial terms and develop some double integral identities. In particular cases it is possible to express the sums in closed form, give some general results, recover some known results in Coffey and produce new identities. 1
unknown title
, 905
"... Abstract. These are seven corrigenda to equations in the Lehmer article in American Mathematical Monthly 92 (1985), pp 449–457, partially reproduced in the Apelblat tables of integrals and series. ..."
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Abstract. These are seven corrigenda to equations in the Lehmer article in American Mathematical Monthly 92 (1985), pp 449–457, partially reproduced in the Apelblat tables of integrals and series.