J.H. McCabe

The symmetric strong moment problem

Sequences of polynomials that occur as denominators in the two point Pade table for two series expansions are considered in the special case when the series coefficients are solutions of a strong symmetric Stieltjes moment problem. The continued fractions whose convergents generate these polynomials as denominators are presented, together with determinant representations for the polynomials and the continued fraction coefficients. The log-normal distribution is used as an example.

On a symmetry in strong distributions

A strong Stieltjes distribution d (t) is called symmetric if it satises the property t ! d ( 2 =t)= ( 2 =t) ! d (t); for t 2(a;b)(0;1);2! 2 Z; and >0: In this article some consequences of symmetry on the moments, the orthogonal L-polynomials and the quadrature formulae associated with the distribution are given. c 1999 Elsevier Science B.V. All rights reserved. MSC: 30E05; 30B70; 41A55

ON THE EXTENSIONS OF SOME CLASSICAL DISTRIBUTIONS

= \are finit ne fo = 0, ±1r +2,..., Suc. h functions have been described as strongdistribution functions because they arise as solutions of strong moment problems (see[1,2]). The distribution is called symmetric if all the odd order moments are zero and iscalled a positive half distribution if all the points of increase are on the positive realaxis.The Hankel determinants are defined by

On the non-normal two-point Pade´ table

Abstract The M -table for two power series expansions, one about the origin and the other about infinity, is generalized to the non-normal case. It is shown that equal entries appear in square blocks, quadrants or half planes. In addition, the continued fractions whose convergents are diagonal or horizontal sequences are constructed. The results are based on a transformation that reduces the study of the two-point table to that of the Pade table.

A class of orthogonal functions given by a three term recurrence formula

The main goal in this manuscript is to present a class of functions satisfying a certain orthogonality property for which there also exists a three term recurrence formula. This class of functions, which can be considered as an extension to the class of symmetric orthogonal polynomials on

On an asymptotic series and corresponding continued fraction for a gamma function ratio

[-1,1]

The Q-D algorithm for transforming series expansions into a corresponding continued fraction: an extension to cope with zero coefficients

, has a complete connection to the orthogonal polynomials on the unit circle. Quadrature rules and other properties based on the zeros of these functions are also considered.

Two-point Padé approximants and the quotient difference algorithm

Abstract Recurrence relations for the coefficients in the asymptotic expansion of a gamma function ratio are derived and a property of these coefficients is proved. The Stieltjes fraction for the series is given and a characteristic of the partial numerators is explained. A connection between the continued fraction and the error of a particular least squares approximation problem is discussed.

Perron fractions : an algorithm for computing the Padé table

An algorithm for deriving a continued fraction that corresponds to two series expansions simultaneously, when there are zero coefficients in one or both series, is given. It is based on using the Q-D algorithm to derive the corresponding fraction for two related series, and then transforming it into the required continued fraction. Two examples are given.

Continued Fractions for the Symmetric Strong Stieltjes Moment Problem

Abstract In a recent paper Sidi considered the two-point Pade approximants to a function for which formal power series expansions at the origin and at infinity are given, and provided determinant representations for them. In comparing his approach with the quotient-difference algorithm, used by McCabe and Murphy on the same problem, certain properties of the coefficients generated by the quotient-difference algorithm are overlooked. It is the purpose of this short note to describe these properties.