Mathematica evidence that Ramanujan kills Baker–Gammel–Wills

摘要

A 1961 conjecture of Baker, Gammel and Wills asserts that if a function f is meromorphic in the unit ball, and analytic at zero, then a subsequence of its diagonal Padé approximants converges uniformly in compact subsets omitting poles. Inasmuch as the denominators of the Padé approximants are complex orthogonal polynomials, and the convergence of sequences of Padé approximants is determined largely by the behaviour of their poles, the conjecture deals with distribution of zeros of complex orthogonal polynomials. In this paper, we present numerical evidence derived using the Mathematica package, that Ramanujan's continued fractionHq(z)=1+qz||1+q2z||1+q3z||1+⋯provides a counterexample, provided q is appropriately chosen on the unit circle.