On orthogonal Laurent polynomials related to the partial sums of power series

Zagorodnyuk, Sergey

Serdica Mathematical Journal, Tome 49 (2023) no. 4, pp. 231-240.

Résumé

Let \(f(z) = \sum_{k=0}^\infty d_k z^k\), \(d_k\in\mathbb{C}\backslash\{ 0 \}\), \(d_0=1\), be a power series with a non-zero radius of convergence \(\rho\): \(0 <\rho \leq +\infty\). Denote by \(f_n(z)\) the \(n\)-th partial sum of \(f\), and \(R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }\), \(R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }\), \(n=0,1,2,\dots\). By the general result of Hendriksen and Van Rossum there exists a unique linear functional \(\mathbf{L}\) on Laurent polynomials, such that \(\mathbf{L}(R_n R_m) = 0\), when \(n\not= m\), while \(\mathbf{L}(R_n^2)\not= 0\), and \(\mathbf{L}(1)=1\). We present an explicit integral representation for \(\mathbf{L}\) in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well.

Détail

Keywords: Laurent polynomials, Maclaurin series, partial sums, orthogonal rational functions, 42C05