Non-unique inverses of certain generalized Stieltjes transforms (Q1892543)

It is shown that the equation \(\int^ \infty_ 0 (s+ t)^ n \phi(t)dt= f(s)\), where \(f\) is a polynomial of degree at most \(n\), has the special solution \(\phi(t)= e^{- t} \sum^ n_{i= 0} a_ i t^ i\) with explicitly given \(a_ i\). The corresponding homogeneous equation has the solutions \(e^{- t} \sum^ \infty_{ i= n+ 1} c_ i L_ i(t)\) with the Laguerre polynomials \(L_ i(t)\) and arbitrary \(c_ i\) with absolutely convergent \(\sum c_ i\), possibly, there exist further solutions. Similar results are given in case of \(d\alpha(t)\) instead of \(\phi(t)dt\).