On the modified convergence of some continued fractions of Rogers- Ramanujan type
In the analytic theory of continued fractions, the method of modified convergence is known for some time. This boils down to replacing the tail of a continued fraction by a fixed or variable number different from zero (that is what happens when a continued fraction is just `cut off'). The author uses for modifier the replacement of the tail starting with \(a_ n\) by \(a_ n + b_{n+1}\), where \(b_{n+1}\) is the next partial numerator. He then proves modified convergence for three different types of \(q\)-continued fractions which can be seen as companions to well known identities of Rogers-Ramanujan and of Gordon. The modified limits are different from the ordinary continued fraction limit in two of the cases and in one of the cases the continued fraction has different limits for the sequences of even and odd approximants and therefore is divergent in the ordinary sense. But the modified continued fraction does converge and the limit is just the mediant of the `odd' and `even' limits.
continued fractions of Rogers-Ramanujan type
scientific article; zbMATH DE number 554398
scientific article