Abelian theorems for a distributional generalized Stieltjes transform (Q1068324)

This paper provides an extension of generalized Stieltjes transform: \[ F(x)=(\lceil a\lceil b/(a+\alpha))(1/x)\int^{\infty}_{0}(t/x)^{\beta}F(a,b,a+\alpha;- t/x)f(t)dt,\quad a=\beta +\eta +1,\quad b=\beta +1; \] to generalized functions (distributions). After defining the spaces \(S_{a,\alpha}(I)\), \(S_{-a,\alpha}(I)\), the author proves that the kernel of the above transform belongs to the space \(S_ a\). \(S_ a\) is the countable union space of the spaces \(S_{a,\alpha}\). An initial value theorem and a final value theorem are also proved in distributional sense for the above generalized Stieltjes transform.