Continued fractions of zero

We have the following continued fraction

in special form.

If we consider q between 0 and 1, we have the following plot

There seems to be several solutions after 0.57… and before 0.9 and many more after 0.9.

0.57614877 is the first one of candidate q’s.

We could find others depending on precision.

Ramanujan considered more general case in terms of parameter a. That is

　　in special form of continued fraction

where

When a=1, q=0.57611879 in this case.

For a>=1, the relation between a and functional value is shown as follows

where the functional value approach zero as parameter a gets larger.