On certain non-unique solutions of the Stieltjes moment problem

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We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ρ₁⁽ʳ⁾(n)=(2rn)! and ρ₂⁽ʳ⁾(n)=[(rn)!]², r=1,2,..., n=0,1,2,..., i.e. we find functions W⁽ʳ⁾₁,₂(x)>0 satisfying ∫₀∞xⁿW⁽ʳ⁾₁,₂(x)dx = ρ₁,₂⁽ʳ⁾(n). It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for r>1 both ρ₁,₂⁽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 ρ₁,₂⁽ʳ⁾(n), such as the product ρ₁⁽ʳ⁾(n)·ρ₂⁽ʳ⁾(n) and [(rn)!]ᵖ, p=3,4,....