Infinite families of designs for Bessel polynomials and associated class numbers

Establish whether there exists an infinite family of designs of degree at least 3 for Bessel polynomials, and, if such a family exists, determine how large the class number of the corresponding parametric solution of the one-dimensional Prouhet–Tarry–Escott problem can be.

Background

The paper connects moments of Bessel polynomials with ideal solutions of the one-dimensional Prouhet–Tarry–Escott problem over imaginary quadratic fields. It notes prior work on weighted designs for the Bessel-polynomial weight and presents an example of a degree-three ideal solution. The authors leave open both the existence of infinite families of Bessel-polynomial designs of degree at least 3 and, conditional on such existence, the possible size of the class number of the associated parametric Prouhet–Tarry–Escott solutions.