A moment problem on some types of hypergroups (Q455470)

There is a classical moment problem: Given a sequence of real numbers \((s_{n})\), find necessary and sufficient conditions for the existence of a measure \(\mu\) on \([0,\infty )\) so that \(s_n=\int_{0}^\infty x^n d\mu (x)\) holds for all \(n\). In analogy to this the authors formulate a similar problem for commutative hypergroups as follows: Given a generalized moment function sequence \((\varphi_{n})\) and a sequence of complex numbers \((m_{n} )\), find necessary and sufficient conditions for the existence of a measure \(\mu\) with compact support so that for every natural number \(n\), \(m_{n}=\int_{K}\varphi_{n} d\mu\). Without establishing these conditions the authors prove that for the case of some special type of one dimensional hypergroups (polynomial hypergroups and Sturm-Liouville hypergroups) if such a measure exists then it is unique.