  We study piecewise polynomial functions $\gamma_k(c)$ that appear in the
asymptotics of averages of the divisor sum in short intervals. Specifically, we
express these polynomials as the inverse Fourier transform of a Hankel
determinant that satisfies a Painlev\'e V equation. We prove that $\gamma_k(c)$
is very smooth at its transition points, and also determine the asymptotics of
$\gamma_k(c)$ in a large neighbourhood of $k=c/2$. Finally, we consider the
coefficients that appear in the asymptotics of elliptic Aliquot cycles.
