The Impact of Stieltjes' Work on Continued Fractions and Orthogonal Polynomials: Additional Material

In the recent new edition of the collected works of T.J. Stieltjes, one of us gave an impression of the impact of Stieltjes' work a century after his death 43]. In this paper we give an update and mention some observations which were missing from 43] and some results which appeared during the last two years and which are directly related to Stieltjes' work. In particular there is a large section on special polynomials which were already considered by Stieltjes but which were rediscovered later, often without the knowledge of Stieltjes' work. 1 Continued fractions and moment problems In 43] an attempt was made to give an updated review of the impact of Stieltjes' work on continued fractions, orthogonal polynomials, and related topics. Here we will mention some extra information which is directly related to Stieltjes' work and which was missing from 43]. First of all we'd like to draw the attention to Kjeldsen's nice historic view on the early history of the moment problem 24], which is recommended reading. In 43] the remark was made (last paragraph of x2.1 and the beginning of x2.2) that`not much work on the Stieltjes moment problem was done after Stieltjes' death' and`nothing new happened until 1920'. What was meant is that the work of Stieltjes was so thorough that he singlehandedly almost completed the theory of the Stieltjes moment problem. Of course there has been work after Stieltjes on the Stieltjes moment problem, e.g., Karlin and McGregor on one hand and Lederman and Reuter on the other hand pointed out the equivalence of Stieltjes' work and the solution of the Chapman-Kolmogorov equations of birth and death processes. Also, the work of Van Vleck, which consists in trying to extend the Stieltjes moment problem to the real line, is quite relevant. Recall that the main object of Stieltjes' work Sur les fractions continues is the S-fraction 1 c 1 z +