Euler seems to be expanding functions into continued fractions of functions. He begins with f(x, y) and finds f(x, y) = A(x, y) + P(y)/f(x, y+1), where A is a linear function. He does it again in the denominator, finding more linear functions B and C so that f(x, y) = B(x, y) + Q(x)/(C(x, y) + f(x+1, y)).
Original Source Citation
Opuscula analytica, Volume 1, pp. 85-120.
Opera Omnia Citation
Series 1, Volume 15, pp.400-434.