On a class of generalized Stieltjes continued fractions

Abstract

With each sequence of real numbers s = {sj}∞j=0 two kinds of continued fractions are associated, — the so-called P-fraction and a generalized Stieltjes fraction that, in the case when s = {sj}∞j=0 is a sequence of moments of a probability measure on +, coincide with the J-fraction and the Stieltjes fraction, respectively. A subclass Hreg of regular sequences is specified for which explicit formulas connecting these two continued fractions are found. For s ∈ Hreg the Darboux transformation of the corresponding generalized Jacobi matrix is calculated in terms of the generalized Stieltjes fraction.