Path generating functions and continued fractions

AbstractThis paper extends Flajolet's (Discrete Math. 32 (1980), 125–161) combinatorial theory of continued fractions by obtaining the generating function for paths between horizontal lines, with arbitrary starting and ending points and weights on the steps. Consequences of the combinatorial arguments used to determine this result are combinatorial proofs for many classical identities involving continued fractions and their convergents, truncations, numerator and denominator polynomials