On the th root of a Stieltjes moment sequence

Stieltjes moment sequences { a n } n = 0 ∞ whose ϰ th roots { a n ϰ } n = 0 ∞ are Stieltjes moment sequences are studied ( ϰ is a fixed integer greater than or equal to 2 ). A formula connecting the closed supports of representing measures of { a n } n = 0 ∞ and { a n ϰ } n = 0 ∞ is established. The relationship between the holes of the supports of these measures is investigated. The set of all pairs ( M , N ) of positive integers for which there exists a Stieltjes moment sequence whose square root is a Stieltjes moment sequence and both of them have representing measures supported on subsets of ( 0 , ∞ ) of cardinality M and N , respectively, is described.