On certain non-unique solutions of the Stieltjes moment problem

Abstract

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form $ρ_{1}^{(r)}(n)=(2rn)!$ and $ρ_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,...$, $n=0,1,2,...$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = ρ_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both $ρ_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing $ρ_{1,2}^{(r)}(n)$, such as the product $ρ_{1}^{(r)}(n)\cdotρ_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,...$.