Explicit Representations of Faber Polynomials form-Cusped Hypocycloids

An Explicit Form for Kerov’s Character Polynomials

Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, Śniady has proved Biane’s conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an...

Lattice Paths and Faber Polynomials

The rth Faber polynomial of the Laurent series f(t) = t + f0 + f1/t + f2/t + · · · is the unique polynomial Fr(u) of degree r in u such that Fr(f) = tr + negative powers of t. We apply Faber polynomials, which were originally used to study univalent functions, to lattice path enumeration.

Derivatives of Faber Polynomials and Markov Inequalities

We study asymptotic behavior of the derivatives of Faber polynomials on a set with corners at the boundary. Our results have applications to the questions of sharpness of Markov inequalities for such sets. In particular, the found asymptotics are related to a general Markov-type inequality of Pommerenke and the associated conjecture of Erdős. We also prove a new bound for Faber polynomials on p...

The Faber Polynomials for Circular Sectors

The Faber polynomials for a region of the complex plane, which are of interest as a basis for polynomial approximation of analytic functions, are determined by a conformai mapping of the complement of that region to the complement of the unit disc. We derive this conformai mapping for a circular sector {;: \z\ < 1, |argz| < i/a}, where a > 1, and obtain a recurrence relation for the coefficient...

Faber Polynomials for Rational Mapping Functions