Gauss's continued fraction

Generalized hypergeometric series are expressed as

where

    for c=1,2,3,…,p+q

We can calculate the elements in numerator and denominator respectively

by multiplying the series p times and q times.

After constructing the vector of i and j, we can use the following formula

   (regular expression)

   in special form of 0218.

It is known that

    for =2,3,4,…

where

We can verify the case of k=2 but not for other cases for k=3,4,5,.. They slightly deviate.