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,. ., i.e. we find functionsW1,2(r) (x)>0 satisfying ∫ 0∞ xnW1,2(r) (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 ρ1(r) (n) · ρ2(r) (n) and [(rn)!]p, p = 3,4,. DMTCS c by the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.