A note on "exotic integrals"

Abstract

We consider Bernoulli measures $μ_p$ 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 $μ_p$-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 $μ_p$ 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.