On the Numerators of the Convergents of the Stieltjes Continued Fractions

Abstract

Introduction.The object of this paper is the study of the numerators of the infinite continued fractions introduced by Stielt jes [1,2].}They are of two types:(a) the "associated" continued fraction:the nth convergent of which will be denoted by Un(z)/<f>n(z)=Kn(z) (n = 0, 1,2,3, );(b) the "corresponding" continued fraction: bi\ b2\ b3\ >J(2) tFW*,-1 N "I -7-, \z \l \z \l the nth convergent of which will be denoted by U"(z)/Vn(z) = W"(z) (n = 0, 1, 2, 3, ).In (a) A,-, c are real constants with A,->0 for i = l, 2,3, ; n(z), 4>n(z) are polynomials of degree n -1 and n respectively^ In (b) >, are real constants, >i>0, b2i+ib2i>0 for i = l, 2, .U2n+(z), V2n+((z) are polynomials of degree (n+e -l) and (n+e) respectively, where e = 0, 1.For our study we shall need certain results from the theory of continued fractions [2 ] and of the so-called moments problem [3].(i) The convergence of the associated and corresponding continued fractions.Here of fundamental importance is Grommer's Selection Theorem [2]:From every sequence of convergents of a continued fraction of type (1) or (2) there may be selected a sub-sequence, which for all non-real z converges to a Stieltjes integral of the form flK(l/(z-u))d\p(u).Here \(/(z) (and hereafter \f/i(z), ip2(z), ) denotes generally a bounded monotonie non-decreasing function with infinitely many points of increase in (-o, ), such that all integrals flaz"d\p(z)=an ("moments") exist (for * Presented to the Society, March 26, 1932; received by the editors April 28, 1932.