Gauss's continued fraction

Suppose that we can say

     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 have the following identity

where we can regard

 and

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