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Mathematica Evidence that Ramanujan Kills Baker-Gammel-Wills

A. Knopfmacher
Organization: Witwatersrand University, Johannesburg, South Africa
D. S. Lubinsky
Organization: Witwatersrand University, Johannesburg, South Africa
Journal / Anthology

Applied Mathematics and Computation
Year: 2002
Volume: 128
Issue: 2-3
Page range: 289-302

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 fraction H_q(z) provides a counterexample, provided q is appropriately chosen on the unit circle.

*Mathematics > Calculus and Analysis > Series
*Mathematics > Number Theory

Padé approximation, continued fractions