Conversion from squared root to continued fraction

Recall that

  in special form

that is the last formula mentioned yesterday.

When y=1/2, it is going to be the expansion of squared root. That is,

Applying gamma-shaped multipliers repeatedly, we get

Applying gamma-shaped multipliers about (x-1) repeatedly, we get

Applying Rule 1(up->front) of signs in 0109, we finally get

   in special form

Recall that

as in 0128. One of the variants can be created in the following way.

Beforehand, applying Rule 1(up->front) of signs in 0109, we can say that

Using the rule above every three terms by inserting one term each,

.

Applying Rule 1(up->front) back, we get

   in special form

where signs of {-.+.+.-} appear in order.