## 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, $${}_r\phi_s$$

Zbl 0102.05902
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