Quasi-definiteness for multiple right-hand jumps

Determine the zero set of the Hankel determinants associated with the Stieltjes-generated moment functional \(\mathcal L_{c_-,c_+,A,B}\) for arbitrary right-hand jump multiset \(A\) with \(|A|\ge 2\) and arbitrary left-hand jump multiset \(B\), thereby giving a complete characterization of quasi-definiteness outside the positive region \(|A|\le 1\).

Background

The paper gives an exact representation of the moment functional generated by a centered Stieltjes derivator in terms of the continuous increments $c_-$, $c_+$, the right-hand jump sizes $A$, and the left-hand jump sizes $B$. It proves that the functional is representable by a positive measure exactly when there are at most one right-hand jump, while two or more right-hand jumps produce endpoint derivative evaluations and generally an indefinite functional.

Although the functional is explicitly known, the paper does not determine for arbitrary $A$ and $B$ when all Hankel determinants are nonzero. The authors specifically identify the complete description of the zero set of these determinants as unresolved, which is necessary for a full classification of quasi-definite but non-positive cases.