Integral Representations and Asymptotics for a Family of Areal Mahler Measures

Abstract

We answer a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$P_n(x_1,\dots ,x_n,u)=\prod _{j=1}^n(1+x_j)+u\prod _{j=1}^n(1-x_j),n\ge 1.$$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for ${\mathrm{m}}_{\mathbb{D}}(P_n)$. These representations yield strict alternating signs for all forward differences of the sequence $({\mathrm{m}}_{\mathbb{D}}(P_n){)}_{n\ge 1}$, as well as an explicit three-term asymptotic expansion as $n\to \infty$.

14 pages