This is obviously a good article written by an expert.
I have three suggestions:
1) The Padé table is now usually the transpose of the table as given in (11), that is the degree of the denominator in fixed in a column.
2) All the identities allowing to compute recursively any sequence of Padé approximants in the table follow from the link between their denominators and numerators with formal orthogonal polynomials as explained in the book: C. Brezinski, Padé-Type Approximation and General Orthogonal Polynomials, ISNM vol. 50, Birkh\"auser Verlag, Basel, 1980.
3) about multi-series, there are as many generalizations as authors who worked on them.
Authors' answer to Reviewer A
I corrected a number of obvious typos in the text. To avoid a potential confusion, I changed some notation. Moreover, there were also some sentences I did not fully understand or I believed I understood but I was not sure. Thus, for clarity purpose I introduced some changes and even added a few sentences. In particular, it was not totally clear what equation 8 meant. Perhaps, in the paragraph concerning Wall contribution, some explanations could be added. The notation used in equation 50 is not defined. I change the wording in the definition and thus possibly the definition itself. Finally, a short sentence concerning Stieltjes function has been added. The extension to multipoint Padé approximants is mentioned. Please check the modifications. Jean Zinn-Justin