On certain non-unique solutions of the Stieltjes moment problem

• Published in: Discrete mathematics and theoretical computer science Vol. 12 no. 2; no. 2; pp. 295 - 306
• Main Authors: Penson, K. A., Blasiak, Pawel, Duchamp, Gérard, Horzela, A., Solomon, A. I.
• Format: Journal Article
• Language: English
• Published: Nancy DMTCS 15.09.2010; Discrete Mathematics & Theoretical Computer Science
• Subjects: [math.math-co] mathematics [math]/combinatorics [math.co]; [math.math-fa] mathematics [math]/functional analysis [math.fa]; Combinatorics; Functional Analysis; Generalized method of moments; Mathematics; Theoretical mathematics
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.507

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${\rho}_{1}^{(r)}(n)=(2rn)!$ and ${\rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,\dots$, $n=0,1,2,\dots$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = {\rho}_{1,2}^{(r)}(n)$. 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 $[(rn)!]^{p}$, $p=3,4,\dots$.