Conjectures Around the Baker – Gammel – Wills Conjecture : Research Problems 97 - 2

The Baker–Gammel-Wills Conjecture states that if a function f is meromorphic in a unit disk D, then there should, at least, exist an infinite subsequence N ⊆ N such that the subsequence of diagonal Padé approximants to f developed at the origin with degrees contained in N converges to f locally uniformly in D/{poles of f }. Despite the fact that this conjecture may well be false in the general form as stated here, it nevertheless has drawn much interest and has influenced research in the theory of Padé approximation in several respects. In the present paper, six new conjectures about the convergence of diagonal Padé approximants are formulated that lead in the same direction as the Baker–Gammel–Wills Conjecture. However, they are more specific and they are based on partial results and theoretical considerations that make it rather probable that these new conjectures hold true.