Non-unique Inverses of Certain Generalized Stieltjes Transforms

Inverses are obtained for generalized Stieltjes transforms with kernel 1/(s + t)p, where p = −n is a negative integer. These inverses are not unique, any two differing by a function having zero moments of order up to n and arbitrary moments of higher order. A sufficient condition for such a function is that it be expressible in a certain way in terms of Laguerre polynomials of order exceeding n, with again a certain degree of arbitrariness.