Conversion from special function to continued fraction

Gamma function is defined as

That is expanded into series as

Since

     in infinite product

We can use the following formula

     (regular expression)

　in special form of

   in special form

Another way is use the following simple rule

  in special form of 0310

      in special case

Its modified rule is

  in special form of 0310

Applying gamma-shaped multipliers, we finally get

    in special form