Hypergeometrics and the cost structure of quadtrees

Flajolet, Philippe; Labelle, Gilbert; Laforest, Louise; Salvy, Bruno

doi:10.1002/rsa.3240070203

Cited by 42 publications

(40 citation statements)

References 14 publications

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“…Of course, the polylogarithm (1.2) reduces to the Riemann zeta function [26], [43], [65] ζ(s) = With each x j = 1, these latter sums (sometimes called "Euler sums"), have been studied previously at various levels of generality [2], [6], [7], [9], [13], [14], [15], [16], [31], [38], [39], [42], [51], [59], the case k = 2 going back to Euler [27]. Recently, Euler sums have arisen in combinatorics (analysis of quad-trees [30], [46] and of lattice reduction algorithms [23]), knot theory [14], [15], [16], [47], and high-energy particle physics [13] (quantum field theory). There is also quite sophisticated work relating polylogarithms and their generalizations to arithmetic and algebraic geometry, and to algebraic K-theory [4], [17], [18], [33], [34], [35], [66], [67], [68].…”

Section: Introductionmentioning

confidence: 99%

Special values of multiple polylogarithms

Borwein

Bradley

Broadhurst

et al. 2000

Trans. Amer. Math. Soc.

356

390

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Abstract. 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 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.

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Section: Introductionmentioning

confidence: 99%

Special values of multiple polylogarithms

Borwein

Bradley

Broadhurst

et al. 2000

Trans. Amer. Math. Soc.

356

390

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“…These expressions obviously show the diversity of the nature of the enumeration problem; see also Flajolet et al [19], Labelle and Laforest [28].…”

Section: Appendix Various Expressions For E[k Nd ]mentioning

confidence: 96%

Maxima in hypercubes

Bai

Devroye

Hwang

et al. 2005

Random Struct. Alg.

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ABSTRACT:We derive a Berry-Esseen bound, essentially of the order of the square of the standard deviation, for the number of maxima in random samples from (0, 1) d . The bound is, although not optimal, the first of its kind for the number of maxima in dimensions higher than two. The proof uses Poisson processes and Stein's method. We also propose a new method for computing the variance and derive an asymptotic expansion. The methods of proof we propose are of some generality and applicable to other regions such as d-dimensional simplex.

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“…In the same way, one verifies P2. In the verification of conditions C1 and L1 we make use of the tail bounds (13) and (14). Let ε > 0.…”

Section: Verifications Of the Conditions In Propositionmentioning

confidence: 99%

On martingale tail sums for the path length in random trees

Sulzbach

2016

Random Struct Algorithms

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For a martingale (Xn) converging almost surely to a random variable X, the sequence (Xn − X) is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko [to appear in Annals of Applied Probability, (2016)] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane-oriented recursive trees.

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Hypergeometrics and the cost structure of quadtrees

Cited by 42 publications

References 14 publications

Special values of multiple polylogarithms

Special values of multiple polylogarithms

Maxima in hypercubes

On martingale tail sums for the path length in random trees

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