Lagrange inversion formula, Laguerre polynomials and the free unitary Brownian motion

The Lagrange Inversion Formula and Divisibility Properties

Wilf stated that the Lagrange inversion formula (LIF) is a remarkable tool for solving certain kinds of functional equations, and at its best it can give explicit formulas where other approaches run into stone walls. Here we present the LIF combinatorially in the form of lattice paths, and apply it to the divisibility property of the coefficients of a formal power series expansion. For the LIF,...

Spectral distribution of the free unitary Brownian motion: another approach

We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any t ∈ (0, 4) a Jordan curve γt around the origin, not intersecting the semi-axis [1,∞[ and whose image under some meromorphic function ht lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and ...

The Planar Tree Lagrange Inversion Formula

Bernstein Polynomials and Brownian Motion

One of the greatest pleasures in mathematics is the surprising connections that often appear between apparently disconnected ideas and theories. Some particularly striking instances exist in the interaction between probability theory and analysis. One of the simplest is the elegant proof of the Weierstrass approximation theorem by S. Bernstein [2]: on the surface, this states that if f : [0, 1]...

Multivariate Lagrange inversion formula and the cycle lemma

We give a multitype extension of the cycle lemma of (Dvoretzky and Motzkin 1947). This allows us to obtain a combinatorial proof of the multivariate Lagrange inversion formula that generalizes the celebrated proof of (Raney 1963) in the univariate case, and its extension in (Chottin 1981) to the two variable case. Until now, only the alternative approach of (Joyal 1981) and (Labelle 1981) via l...