Conversion from some integral to continued fraction

Complete elliptic integrals of first kind and second kind are defined as

First kind:

   for 0<x<1

Second kind:

    for 0<x<1

Their expansions into series are

and

The first kind can be converted into continued fraction by the formula

                       in special form

in 0215.

Regarding  and manipulating the c₀,

   in special form

or applying gamma-shaped multipliers,

The second kind can be converted into continued fraction by the formula

  (regular expression)

   in special form

by rules of signs(both->next, down->front&next, down->front&next,…)

where