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 ${\rho }_1^{(r)}(n)=(2rn)!$ and ${\rho }_2^{(r)}(n)=[(rn)!{]}^2$, $r=1,2,...$, $n=0,1,2,...$, \textit{i.e.} we find functions $W_{1,2}^{(r)}(x)>0$ satisfying ${\int }_0^{\infty }{x}^n{W}_{1,2}^{(r)}(x)dx={\rho }_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${\rho }_{1,2}^{(r)}(n)$ 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 ${\rho }_{1,2}^{(r)}(n)$, such as the product ${\rho }_1^{(r)}(n)\cdot {\rho }_2^{(r)}(n)$ and $[(rn)!{]}^p$, $p=3,4,...$.

References in corpus (1)

• Coherent States from Combinatorial Sequences