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Application of Faber polynomials in proving combinatorial identities. (English. Ukrainian original) Zbl 1428.05032

Ukr. Math. J. 70, No. 2, 165-181 (2018); translation from Ukr. Mat. Zh. 70, No. 2, 151-164 (2018).
Suppose \(K\) is a compact subset of \(\mathbb C\) containing at least \(2\) points. If the complement of \(K\) in the Riemann sphere \(\overline{\mathbb C}=\mathbb C\cup\{\infty\}\) is simply connected, then the Riemann mapping theorem guarantees that there exists a unique biholomorphic map \(\Phi:\overline{\mathbb C}\setminus K\to\{w\in\overline{\mathbb C}:|w|>1\}\) such that \(\Phi(\infty)=\infty\) and \(\Phi^\prime(\infty)=1\). By extracting the coefficients of the powers of the Laurent series expansions of \(\Phi\) and \(\Phi^{-1}\) about \(\infty\), we obtain two sequences of polynomials depending only on the set \(K\). These are called Faber polynomials. The main result of the present paper is a new general identity among the coefficients of these Faber polynomials. The paper then proceeds to apply this result in four specific cases in order to derive combinatorial identities. More precisely, the specific choices of \(K\) that the paper considers are:

(i) the \(n\)-lemniscate \(\{z\in\mathbb C:|z^n-1|\leq 1\}\);
(ii) the \(n\)-ray star \(\bigcup_{j=1}^n\{z\in\mathbb C:0\leq|z|\leq 4^{1/n},\arg(z)=e^{2\pi i(j-1)/n}\}\);
(iii) the closure of the domain bounded by the \(n\)-hypocycloid \(\{e^{i\theta}+e^{-n\theta}/n:0\leq \theta<2\pi\}\);
(iv) the Lambert drop, which is defined to be the closure of the domain bounded by \(\{e^{\cos\theta+i(\theta-\sin\theta)}:0\leq\theta<2\pi\}\).
For example, one specific combinatorial identity that follows from the main theorem of the paper and an analysis of the Lambert drop Faber polynomials is \[\sum_{k=0}^{m-\ell}(-1)^{m-k-\ell}{m-\frac{\ell}{k}}(k+\ell+1)^{k-1}(k+\ell)^{m-k-\ell-1}=-\frac{m-\ell-1}{m(\ell+1)},\] which holds for \(m\geq \ell\).

MSC:

05A19 Combinatorial identities, bijective combinatorics
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References:

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