Conversion from binomial expansion to continued fraction

We all know generalized binomial theorem

   for

or

   for

When y=1 and we set r=y

   for

That is

Arranging the terms, we can regard it as

Now we apply one of the formulas from series to continued fraction. The formula is

   (regular expression)

   in special form

as in 0218. We set i={1,yx,(y-1)x,(y-2)x,(y-3)x,. . .} and j={1,1,2,3,4,. . .}.

Then we have

Thus

or

  in special form

Recall

in 0122. Now a₁=x*y and b₁=1.