You can find this expansion in the book *Lorentzen L. & Waadeland H. Continued fractions with applications North-Holland Publishing Co., 1992* (formula (4.3.10)). As I understand, this [document][1] is a more recent version of the appendix of this book. Here the desired formula has the number (3.3.10).



  [1]: https://link.springer.com/content/pdf/bbm%3A978-94-91216-37-4%2F1