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Differential calculus on the Faber polynomials. (English) Zbl 1163.30301

From the introduction: We show how the methods introduced in [Bull. Sci. Math. 126, No. 5, 343–367 (2002; Zbl 1010.33006)] and [A. Bouali, ibid. 130, No. 1, 49-70 (2006; Zbl 1094.30010)] allow to do differential calculus on the manifold of coefficients of univalent functions. The Faber polynomials \((F_k)_{k\geq 1}\) are given by the identity
\[ 1+b_1w+ b_2w^2+\cdots+ b_kw^k+\cdots= \exp\Bigg(-\sum_{k=1}^{+\infty} \frac{F_k(b_1,b_2,\dots,b_k)}{k} w^k\Bigg). \]
The polynomials \((G_m)_{m\geq 1}\) and \((K_n^p)_{n\geq 1}\), \(p\in\mathbb Z\), are given by
\[ \begin{aligned} \frac{1}{1+b_1w+b_2w^2+\cdots+b_kw^k+\cdots} &= 1+ \sum_{m=1}^{+\infty} G_m(b_1,b_2,\dots,b_m)w^m,\\ (1+b_1w+b_2w^2+\cdots+b_kw^k+\cdots)^p &= 1+ \sum_{n\geq 1} K_n^p(b_1,b_2,\dots,b_n)w^n, \end{aligned} \]
then \(G_m= K_m^{-1}\) and \(K_m^1=b_m\).
The object of this note is to prove that the polynomials \((K_n^p)\) are all obtained as partial derivatives of the Faber polynomials and show how some of the recursion formulae on the polynomials are related to elementary differential calculus on \({\mathcal M}\). This is a step towards the classification of Faber type polynomials. In the last section, we give the example of the conformal map from the exterior of the unit disk onto the exterior of \([-2,+2]\). This shows how to introduce nontrivial second-order differential operators on the manifold \({\mathcal M}\).

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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