On the numerators of the convergents of the Stieltjes continued fractions

Chain sequences and orthogonal polynomials

“…and the above results also follow from the classical theory of continued fractions (for example, see [10] and the remarks in [5]). In view of this, it seems natural to attempt a study of the properties of the polynomials, Pn(x), as determined by the coefficients, cn and Xn.…”

“…(Thus the P*'(x) are the "numerator polynomials" [5] of the P[*-1)(x).) (a(*)| ¿,(fc)) denotes the true interval of orthogonality of {P£>(x)} ; (a(0), ö(0)) = (a,b).…”

First return probabilities of birth and death chains and associated orthogonal polynomials

“…Note that Theorem 2.1 gives a new characterization of the measure of orthogonality corresponding to the first associated orthogonal polynomials, which is different from the representations commonly used in the literature (see e.g. Sherman [15] or Grosjean [9]). For example, it is shown in Van Assche [19] that the Stieltjes transform of the orthogonality measure corresponding to the first associated orthogonal polynomials is given by the equation (2.6) and ψ (1) could be determined from ψ by the Stieltjes inversion formula.…”

Hamburger moment problems and orthogonal polynomials

“…and conclude that the corresponding Hamburger moment problem is indeterminate. But Sherman [13] has noted that if the Hamburger moment problem for (1.1) is determined, so is the moment problem for the corresponding numerator polynomials (of (2.15)); hence by induction, the moment problem for the numerator polynomials of order j is determined. It thus follows that the moment problem for (1.1) must be indeterminate.…”