  There is currently a growing interest in understanding which lattice
simplices have unimodal local $h^\ast$-polynomials (sometimes called box
polynomials); specifically in light of their potential applications to
unimodality questions for Ehrhart $h^\ast$-polynomials. In this note, we
compute a general form for the local $h^\ast$-polynomial of a well-studied
family of lattice simplices whose associated toric varieties are weighted
projective spaces. We then apply this formula to prove that certain such
lattice simplices, whose combinatorics are naturally encoded using common
systems of numeration, all have real-rooted, and thus unimodal, local
$h^\ast$-polynomials. As a consequence, we discover a new restricted Eulerian
polynomial that is real-rooted, symmetric, and admits intriguing number
theoretic properties.
