Central Binomial Sums, Multiple Clausen Values, and Zeta Values

Borwein, Jonathan M.; Broadhurst, David J.; Kamnitzer, Joel

doi:10.1080/10586458.2001.10504426

Cited by 75 publications

(107 citation statements)

References 9 publications

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“…In ref. [50], the multiple Glaishers and multiple Clausen functions were introduced as the real and imaginary parts of generalized polylogarithms of complex unit argument. In ref.…”

Section: Discussionmentioning

confidence: 99%

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Kalmykov¹,

Ward²,

Yost³

2007

J. High Energy Phys.

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“…In ref. [50], the multiple Glaishers and multiple Clausen functions were introduced as the real and imaginary parts of generalized polylogarithms of complex unit argument. In ref.…”

Section: Discussionmentioning

confidence: 99%

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Kalmykov¹,

Ward²,

Yost³

2007

J. High Energy Phys.

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“…In general, irreducibility means that a polylogarithmic value is not expressible as a sum of products of lower dimensional polylogarithmic values. Conjectures concerning the number of irreducibles at each depth can be found in [8] in the case of Clausen and Glaisher values, and in [52] for polylogarithms in general. It is worth emphasizing that all such conjectures are necessarily experimental -they would collapse in the unlikely event that it were shown that some odd ζ-value, say ζ(11), was rational!…”

Section: Lsmentioning

confidence: 99%

“…obtained in [8]. Similarly, alternating binomial sums are related to log-sinh integrals at 2 log ρ where ρ = 1+ √ 5 2 is the golden mean.…”

Section: Lsmentioning

confidence: 99%

“…n (τ ) can be done in terms of Nielsen polylogarithms with argument e iτ , which split into Clausen and Glaisher functions at τ according to (8) and (9). More specifically, the following reduction formula, valid for 0 τ 2π and n − k 2, is derived in [13]: This formula may be used recursively to automatically express the log-sine values Ls Related values at π/3 can also be found in [19].…”

Section: Lsmentioning

confidence: 99%

“…Because their real and imaginary parts satisfy various relations and reductions, studied in [8], the evaluations for log-sine integrals Ls…”

mentioning

confidence: 99%

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Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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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 log-sine integrals we provide systematic evaluations of various higher and multiple Mahler measures.The evaluations in terms of log-sine integrals become particularly useful in light of the fact that log-sine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for log-sine 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 n-step walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures.This work would be impossible without very extensive symbolic and numeric computations. It also makes frequent use of the new NIST Handbook of Mathematical Functions and similar tools. Our intention is to show off the interplay between numeric and symbolic computing while exploring the three mathematical topics in the title.

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Electrostatic energies and forces computed without explicit interparticle interactions: A linear time complexity formulation

Petrella¹,

Karplus²

2005

J Comput Chem

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A rapid method for the calculation of the electrostatic energy of a system without a cutoff is described in which the computational time grows linearly with the number of particles or charges. The inverse of the distance is approximated as a polynomial, which is then transformed into a function whose terms involve individual particles, instead of particle pairs, by a partitioning of the double sum. In this way, the electrostatic energy that is determined by the interparticle interactions is obtained without explicit calculation of these interactions. For systems of positive charges positioned on a face-centered cubic lattice, the calculation of the energy by the new method is shown to be faster than the calculation of the exact energy, in many cases by an order of magnitude, and to be accurate to within 1-2%. The application of this method to increase the accuracy of conventional truncation-based calculations in condensed-phase systems is also demonstrated by combining the approximated long-range electrostatic interactions with the exact short-range interactions in a "hybrid" calculation. For a 20-A sphere of water molecules, the forces are shown to be six times as accurate using this hybrid method as those calculated with conventional truncation of the electrostatic energy function at 12 A. This is accomplished with a slight increase in speed, and with a sevenfold increase in speed relative to the exact all-pair calculation. Structures minimized with the hybrid function are shown to be closer to structures minimized with an exact all-pair electrostatic energy function than are those minimized with a conventional 13-A cutoff-based electrostatic energy function. Comparison of the energies and forces calculated with the exact method illustrate that the absolute errors obtained with standard truncation can be very large. The extension of the current method to other pairwise functions as well as to multibody functions, is described.

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Central Binomial Sums, Multiple Clausen Values, and Zeta Values

Cited by 75 publications

References 9 publications

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

Mahler measures, short walks and log-sine integrals

Electrostatic energies and forces computed without explicit interparticle interactions: A linear time complexity formulation

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