Conversion from special function to continued fraction

Bessel function of the first kind of order n is expressed as

in infinite products. Here,  are positive roots of Jn(x)=0.

We can use the following formula

     (regular expression)

   in special form

as in 0220 but it is difficult to find the roots or it take a while to find out the roots.

Another way is expand

in series. We can use the following formula

  (regular expression)

   in special form

Hypergeometric function is defined and expanded as

We can use the following formula

   (regular expression)

   in special form of 0218.

Regarding

we get

in special form of continued fraction

When x=1, in particular,

                when c>0, c-a-b>0, c>a, and c>b.