A note on "exotic integrals"

TL;DR

This paper derives explicit formulas for Bernoulli measure integrals on [0,1], connecting polynomial, Fourier, and special functions, revealing new relationships involving Stirling numbers and polylogarithms.

Contribution

It provides explicit expressions for Bernoulli measure integrals, including polynomial and Fourier coefficients, using Hessenberg and Pascal matrices, and explores properties of related entire functions.

Findings

Explicit formulas for polynomial integrals in terms of Hessenberg matrices.

Closed-form expressions for Fourier coefficients in Legendre basis.

Identification of special entire functions with connections to Stirling numbers and polylogarithms.

Abstract

We consider Bernoulli measures $\mu_p$ on the interval $[0,1]$. For the standard Lebesgue measure the digits $0$ and $1$ in the binary representation of real numbers appear with an equal probability $1/2$. For the Bernoulli measures, the digits $0$ and $1$ appear with probabilities $p$ and $1-p$, respectively. We provide explicit expressions for various $\mu_p$-integrals. In particular, integrals of polynomials are expressed in terms of the determinants of special Hessenberg matrices, which, in turn, are constructed from the Pascal matrices of binomial coefficients. This allows us to find closed-form expressions for the Fourier coefficients of $\mu_p$ in the Legendre polynomial basis. At the same time, the trigonometric Fourier coefficients are values of some special entire function, which admits an explicit infinite product expansion and satisfies interesting properties, including connections with the Stirling numbers and the polylogarithm.