Gauss's continued fraction

Hypergeometric series are expressed as

where p is the number of kinds of rising factorials on the numerator and q is that on denominator as in generalized hypergeometric series.

The rising factorial is

For these, we can use the following formula

   (regular expression)

   in special form of 0218.

For the first one,

Regarding

We get the continued fraction

Regarding in the same way

we get the following continued fraction

For the last one and others we can do the same thing as what we did yesterday.