On a class of generalized Stieltjes continued fractions (Q2822219)

Let \(\{ s_j\}_{j=0}^\infty\) be the moment sequence of a nonnegative measure \(\sigma\) on \(\mathbb R\). If the moment problem is nondegenerate and \(\operatorname{supp}\sigma\) is contained in a finite interval of \(\mathbb R_+\), there are two well-known constructions of the continued fraction expansion for the Stieltjes transform of \(\sigma\): the J-fraction and the Stieltjes fraction. The authors propose extensions of these constructions to general real sequences. A subclass of regular sequences is found, for which there are explicit formulas connecting these two continued fractions. For this case, the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.