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Rogers-Ramanujan and the Baker-Gammel-Wills (Padé) conjecture. (English) Zbl 1068.41033

This quite interesting paper addresses an old conjecture attributed to Baker-Gammel-Wills [G. A. Baker, J. L. Gammel and J. G. Wills, J. Math. Anal. Appl. 2, 405–418 (1961; Zbl 0102.05902)], that for any function \(f\) meromorphic on the unit ball and analytic at the origin, there is an infinite subsequence of the diagonal sequence \([n/n],\;n\geq 1\), of Padé approximants that converges uniformly on all compact subsets of the unit ball, omitting the poles of \(f\).
The main results consists of a counter example generated by the Rogers-Ramanujan function \[ G_q(z)=\sum_{j=0}^{\infty}\, {q^{j^2}\over (1-q)(1-q^2)\cdots (1-q^j)}, \] and runs as follows.
Define \(H_q(z)=G_q(z)/G_q(qz)\) with \(q:=\exp{(2\pi i \tau)}\), where \(\tau= 2/(99+\sqrt{5})\). Then \(H_q\) is meromorphic in the unit ball and analytic at the origin, but: there does not exist a subsequence of \([n/n],\;n\geq 1\), or of \([n+1/n],\;n\geq 1\), that converges on all compact subsets of \(\{z: | z| <0.46\}\), omitting the poles of \(H_q\).
In view of Worpitzky’s theorem (no matter the value of \(q\), even the full sequence of diagonal approximants converges uniformly on \(\{z:| z| <1/4\}\), the solution of the conjecture immediately leads to other convergence problems and also to problems concerning the structure of the zero-set of \(H_q\). The full proof is rather intricate and it took the author a number of years of perseverence (running side to side to all his other research accomplishments); he is to be complimented for his clear exposé.

MSC:

41A21 Padé approximation
30E10 Approximation in the complex plane
11J70 Continued fractions and generalizations
30B70 Continued fractions; complex-analytic aspects
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)

Citations:

Zbl 0102.05902
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