Gauss's continued fraction

We continue to start with

     for i=1,2,3,4,…

where fi is a function and ki is a constant and x is a variable that is used in function fi.

  or

This can be recursively expanded as

 or  for i=1

We now consider the following two identities

and

and use them by turns. That is

, and +1 to both of a and b

, and +1 to b only

, and +1 to both of a and b

, and +1 to b only

, and +1 to both of a and b

, and +1 to b only

 .

 .

 .

For

we can regard ki’s as

where a pattern on numerator appears alternately. That is, we can expand it as

or

in special form

Applying gamma-shaped multipliers repeatedly, we get

or

   in special form

Thus

   in special form

Starting from the second identity, we also can get

or

in special form

Applying gamma-shaped multipliers repeatedly, we get

or

   in special form

Thus

   in special form