## 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.)

### Citations:

Zbl 1010.33006; Zbl 1094.30010
Full Text:

### References:

 [1] Airault, H.; Malliavin, P., Unitarizing probability measures for representations of Virasoro algebra, J. math. pures appl., 80, 6, 627-667, (2001) · Zbl 1032.58021 [2] Airault, H.; Ren, J., An algebra of differential operators and generating functions on the set of univalent functions, Bull. sci. math., 126, 5, 343-367, (2002) · Zbl 1010.33006 [3] A. Bouali, Faber polynomials, Cayley-Hamilton equation and Newton symmetric functions, Bull. Sci. Math. (2005) · Zbl 1094.30010 [4] A. Bouali, On the Faber polynomials of a rectangle, preprint, 2005 [5] Faber, G., Uber polynomische entwicklungen, Math. ann., 57, 385-408, (1903) · JFM 34.0430.01 [6] Feller, W., An introduction to probability theory and its applications, vol. 1, (1968), John Wiley · Zbl 0155.23101 [7] Kirillov, A.A., Geometric approach to discrete series of unireps for Virasoro, J. math. pures appl., 77, 735-746, (1998) · Zbl 0922.58078 [8] Montel, P., Leçons sur LES séries de polynômes à une variable complexe, Collection de monographies sur la théorie des fonctions, (1910), Gauthier-Villars Paris · JFM 41.0277.01 [9] Neretin, Y.A., Representations of Virasoro and affine Lie algebras, (), 157-225 · Zbl 0805.17018 [10] Pritsker, I.E., Derivatives of Faber polynomials and Markov inequalities, J. approx. theory, 118, 163-174, (2002) · Zbl 1374.30012 [11] Schaeffer, A.C.; Spencer, D.C., Coefficient regions for schlicht function, Colloquium publications, vol. 35, (1950), American Math. Soc. · Zbl 0066.05701 [12] Schiffer, M., Faber polynomials in the theory of univalent functions, Bull. amer. soc., 54, 503-517, (1948) · Zbl 0033.36301 [13] Schur, I., Identities in the theory of power series, Amer. J. math., 69, 14-26, (1947) · Zbl 0034.01103
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