Combinatorial aspects of continued fractions

We show that the universal continued fraction of the Stieltjes-Jacobi type is equivalent to the characteristic series of labelled paths in the plane . The equivalence holds in the set of series in non-commutative indeterminates . Using it, we derive direct combinatorial proofs of continued fraction expansions for series involving known combinatorial quantities : the Catalan numbers, the Bell and Stirling numbers, the tangent and secant numbers, the Euler and Eulerian numbers . . . . We also show combinatorial interpretations for the coefficients of the elliptic functions, the coefficients of inverses of the Tchebycheff, Charlier, Hermite, Laguerre and Meixner polynomials . Other applications include cycles of binomial coefficients and inversion formulae . Most of the proofs follow from direct geometrical correspondences between objects .