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Combinatorial aspects of continued fractions. (English) Zbl 0445.05014
Author’s summary: 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 \(\dots\). 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.
Reviewer: R. C. Read

MSC:
05A15 Exact enumeration problems, generating functions
05A10 Factorials, binomial coefficients, combinatorial functions
30B70 Continued fractions; complex-analytic aspects
11A55 Continued fractions
11B65 Binomial coefficients; factorials; \(q\)-identities
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
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