Gauss's continued fraction

We know that

   in special form

as mentioned the day before yesterday.

We also know that

 or  whenever we have a=0.

That is

or

When b=b-1

Applying gamma-shaped multiplier of b-1 at the beginning, we can get

　　in special form

We also know that

   in special form

as mentioned yesterday.

We also know that

 whenever we have a=0.

That is

or

When c=c-1

Applying gamma-shaped multiplier of c-1 at the beginning, we can get

　　in special form