On the Rogers--Ramanujan Periodic Continued Fraction.

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction <MATH> 1+\frac{qz}{1+\frac{q^2z}{1+\cdots}} </MATH> are studied for <MATH>q=\exp (2\pi i\tau)</MATH>, where <MATH>\tau</MATH> is a rational number. It is shown that the function <MATH>H_q</MATH> to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational <MATH>\tau </MATH>, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function <MATH>H_q</MATH> does not exceed one half of its genus. [ABSTRACT FROM AUTHOR]