Logarithmic integrals, zeta values, and tiered binomial coefficients

TL;DR

This paper explores the relationships between logarithmic integrals, zeta values, and tiered binomial coefficients, revealing new properties and applications to the analysis of Quicksort algorithm comparisons.

Contribution

It introduces tiered binomial coefficients, establishes their properties, and applies these findings to analyze the moments and cumulants of Quicksort's limit distribution.

Findings

Logarithmic integrals can be expressed using tiered binomial coefficients and zeta values.

Properties of tiered binomial coefficients involve binomial transforms and multiple zeta values.

Reproves the rational polynomial nature of Quicksort moments and presents a new expression for its cumulants.

Abstract

We study logarithmic integrals of the form $\int_0^1 x^i\ln^n(x)\ln^m(1-x)dx$. They are expressed as a rational linear combination of certain rational numbers $(n,m)_i$, which we call tiered binomial coefficients, and products of the zeta values $\zeta(2)$, $\zeta(3)$,\dots. Various properties of the tiered binomial coefficients are established. They involve, amongst others, the binomial transform, truncated multiple zeta and multiple zeta star values, as well as special functions. As an application we discuss the limit law of the number of comparisons of the Quicksort algorithm: we reprove that the moments of the limit law are rational polynomials in the zeta values. A novel expression for the cumulants of the Quicksort limit is also presented.